2.1

We denote by $\mathbf {X}$ an infinite alphabet $x_1,x_2,\dots $ and by $\operatorname {\mathrm {Sym}}[\mathbf {X}]$ the ring of symmetric functions in $\mathbf {X}$ . The field or ring of coefficients $\mathscr {K}\supseteq \mathbb Q$ will depend on the context. For any $k\geq 1$ , we denote by $\overline {\mathbf {X}}_k$ the finite alphabet $x_1,x_2,\dots , x_k$ and by ${\mathbf {X}}_k$ the infinite alphabet $x_{k+1},x_{k+2},\dots $ . $\operatorname {\mathrm {Sym}}[{\mathbf {X}}_k]$ will denote the ring of symmetric functions in ${\mathbf {X}}_k$ . Furthermore, for any $1\leq k\leq m$ , we denote by $\overline {\mathbf {X}}_{[k,m]}$ the finite alphabet $x_k,\dots , x_m$ . As usual, we denote by $h_n[\mathbf {X}]$ (or $h_n[\overline {\mathbf {X}}_k]$ or $h_n[\mathbf {X}_k]$ or $h_n[\overline {\mathbf {X}}_{[k,m]}]$ ) the n-th complete symmetric functions (or polynomials) in the indicated alphabet, by $p_n[\mathbf {X}]$ (or $p_n[\overline {\mathbf {X}}_k]$ or $p_n[\mathbf {X}_k]$ or $p_n[\overline {\mathbf {X}}_{[k,m]}]$ ) the n-th power sum symmetric functions (or polynomials), and by $e_n[\mathbf {X}]$ (or $e_n[\overline {\mathbf {X}}_k]$ or $e_n[\mathbf {X}_k]$ or $e_n[\overline {\mathbf {X}}_{[k,m]}]$ ) the n-th elementary symmetric functions (or polynomials). The symmetric function $p_1[\mathbf {X}]=h_1[\mathbf {X}]=e_1[\mathbf {X}]$ is also denoted by $\mathbf {X}=x_1+x_2+\cdots $ . For a partition $\lambda $ , $m_{\lambda }[\mathbf {X}]$ (or $m_{\lambda }[\overline {\mathbf {X}}_k]$ or $m_{\lambda }[\mathbf {X}_k]$ ) denotes the monomial symmetric function (or polynomial) in the indicated alphabet.

2.2

Any action of the monoid $(\mathbb Z_{>0},\cdot )$ on the ring $\mathscr {K}$ extends to a canonical action by $\mathbb Q$ -algebra morphisms on $\operatorname {\mathrm {Sym}}[\mathbf {X}]$ . The morphism corresponding to the action of $n\in \mathbb Q_{>0}$ is denoted by $\mathfrak {p}_n$ and is defined by

$$ \begin{align*} \mathfrak{p}_n\cdot p_k[\mathbf{X}]=p_{nk}[\mathbf{X}], \quad k\geq 1. \end{align*} $$

In our context, $\mathscr {K}$ will be a ring of polynomials or a field of fractions generated by some finite set of parameters, the action of $(\mathbb Z_{>0},\cdot )$ on $\mathscr {K}$ is $\mathbb Q$ -linear, and $\mathfrak {p}_n$ acts on parameters by raising them to the n-th power. For example, if $\mathscr {K}=\mathbb Q(\textbf {t},\textbf {q})$ , then $\mathfrak {p}_n\cdot \textbf {t}=\textbf {t}^n, ~\mathfrak {p}_n\cdot \textbf {q}=\textbf {q}^n$ .

Let R be a ring with an action of $(\mathbb Z_{>0},\cdot )$ by ring morphisms. Any ring morphism $\varphi : \operatorname {\mathrm {Sym}}[\mathbf {X}]\to R$ that is compatible with the action of $(\mathbb Z_{>0},\cdot )$ is uniquely determined by the image of $p_1[\mathbf {X}]=\mathbf {X}$ . The image of $F[\mathbf {X}]\in \operatorname {\mathrm {Sym}}[\mathbf {X}]$ through $\varphi $ is usually denoted by $F[\varphi (\mathbf {X})]$ and called the plethystic evaluation (or substitution) of F at $\varphi (\mathbf {X})$ .

The plethystic exponential $\textrm {Exp}[\mathbf {X}]$ is defined as

$$ \begin{align*} \textrm{Exp}[\mathbf{X}]=\sum_{n=0}^{\infty}h_n[\mathbf{X}]=\exp\left(\sum_{n=1}^{\infty} \frac{p_n[\mathbf{X}]}{n}\right). \end{align*} $$

2.3

We will use some symmetric polynomials that are related to the complete homogeneous symmetric functions via plethystic substitution. More precisely, let $h_n[(1-\textbf {t})\overline {\mathbf {X}}_k]$ be the symmetric polynomial obtained from the symmetric function $h_n[(1-\textbf {t})\mathbf {X}]$ by specializing to $0$ the elements of the alphabet ${\mathbf {X}}_k$ . The corresponding notation applies to $h_n[(\textbf {t}-1)\mathbf {X}]$ and other plethystic substitutions.

For example, from the Newton identities for the complete homogeneous symmetric functions, we can easily see that (for $x=x_1$ )

(2.1) $$ \begin{align} h_n[(1-\textbf{t})x]=(1-\textbf{t})x^n, \quad h_n[(\textbf{t}-1)x]=\textbf{t}^{n-1}(\textbf{t}-1)x^n, n\geq 1, \end{align} $$

and $h_0[(1-\textbf {t})x]=h_0[(\textbf {t}-1)x]=1$ . The classical convolution formula $\displaystyle h_n[\mathbf {X}]=\sum _{i+j=n} h_i[\mathbf {X}] h_j[\mathbf {X}]$ has the following counterparts

(2.2) $$ \begin{align} h_n[(1-\textbf{t})\overline{\mathbf{X}}_k]=\sum_{i+j=n} h_i[(1-\textbf{t})\overline{\mathbf{X}}_{k-1}]h_j[(1-\textbf{t})x_k] \end{align} $$

and

(2.3) $$ \begin{align} h_n[(1-\textbf{t})\overline{\mathbf{X}}_{k-1}]=\sum_{i+j=n} h_i[(1-\textbf{t})\overline{\mathbf{X}}_{k}]h_j[(\textbf{t}-1)x_k]. \end{align} $$

We will make use of such formulas applied to similar situations—for example, with $\overline {\mathbf {X}}_{[k+1,m]}$ , $m>k$ , replacing $\overline {\mathbf {X}}_{k}$ in the formulas above.

2.4

For any $k\geq 1$ , let $\mathscr {P}_k=\mathbb Q(\textbf {t},\textbf {q})[x_1^{\pm 1},\dots ,x_k^{\pm 1}]$ be the ring of Laurent polynomials in the variables $x_1,\dots , x_k$ . The symmetric group $S_k$ acts on $\mathscr {P}_k$ by permuting the variables. We denote by $s_i$ the simple transposition that interchanges $x_i$ and $x_{i+1}$ and is fixing all the other variables. The polynomial subrings

$$ \begin{align*} \mathscr{P}_k^+=\mathbb Q(\textbf{t},\textbf{q})[x_1,\dots,x_k]\quad \text{and}\quad \mathscr{P}_k^-=\mathbb Q (\textbf{t}, \textbf{q})[x_1^{-1},\dots,x_k^{-1}] \end{align*} $$

are stable under the action of $S_k$ .

2.5

Let

$$ \begin{align*} \pi_k:\mathscr{P}^+_k\rightarrow \mathscr{P}^+_{k-1},\quad \text{and } \quad \iota_k:\mathscr{P}_{k-1}^{+}\rightarrow \mathscr{P}_{k}^{+} \end{align*} $$

be the evaluation morphism that maps $x_{k}$ to $0$ and, respectively, the canonical inclusion. The definitions and facts that we discuss below have straightforward analogues associated to the sequence of rings $\mathscr {P}_k^-$ , $k\geq 1$ . To avoid introducing additional notation, we will use $\pi _k$ and $\iota _k$ for the corresponding morphisms between the rings $\mathscr {P}_k^-$ and $\mathscr {P}_{k-1}^-$ (to be clear, in this case, the evaluation morphism $\pi _k$ maps $x_k^{-1}$ to $0$ ), with the hope that the necessary distinction will be clear from the context (in fact, the objects associated with $\mathscr {P}_k^-$ will appear only in §5). We adopt the same convention for all the other morphisms considered in this subsection: $\iota _{n,k}$ , $\Pi _k$ , $I_k$ , $J_k$ , J.

The rings $\mathscr {P}_k^{\pm }$ , $k\geq 1$ form an graded inverse system. We will use the notation $\mathscr {P}_{\infty }^{\pm }$ for the graded inverse limit ring $\displaystyle \lim _{\longleftarrow }\mathscr {P}_k^{\pm }$ . The graded inverse limit rings are sometimes referred to in the literature as the rings of formal polynomials in the variables $x_i^{\pm }$ , $i\geq 1$ . We denote by $\Pi _k: \displaystyle \lim _{\longleftarrow }\mathscr {P}_k^{\pm }\to \mathscr {P}_k^{\pm }$ the canonical morphism.

The inductive limit $\displaystyle \lim _{{\longrightarrow }} \mathscr {P}_k^+$ is canonically isomorphic with the ring $\mathbb {Q}(\textbf {t},\textbf {q})[x_1,x_2,\dots ]$ of polynomials in infinitely many variables (and similarly for $\displaystyle \lim _{{\longrightarrow }} \mathscr {P}_k^-$ ). As with the projective limit, the rings $\displaystyle \lim _{\substack {\longrightarrow \\ k\geq n}} \mathscr {P}_k^{\pm }$ and $\displaystyle \lim _{{\longrightarrow }} \mathscr {P}_k^{\pm }$ are canonically isomorphic. We denote by $I_k: \displaystyle \mathscr {P}_k^{\pm }\to \lim _{{\longrightarrow }} \mathscr {P}_k^{\pm }$ the canonical morphism.

The following diagram is commutative:

where each horizontal map represents the identity map. For a fixed $n\geq 1$ , denote by $\iota _{n,k}: \mathscr {P}_n^{\pm }\to \mathscr {P}_k^{\pm }$ , $n\leq k$ , the canonical inclusion. The sequence of maps $\iota _{n,k}: \mathscr {P}_n^{\pm }\to \mathscr {P}_k^{\pm }$ , $k\geq n$ is compatible with the structure maps $\pi _k$ . Therefore, they induce a morphism

$$ \begin{align*}\mathscr{P}_n^{\pm}\to \lim_{\substack{\longrightarrow \\ k\geq n}} \mathscr{P}_k^{\pm}\cong \displaystyle \lim_{{\longrightarrow }} \mathscr{P}_k^{\pm}. \end{align*} $$

Furthermore, these maps are compatible with the structure maps $\iota _n$ and therefore induce a morphism

$$ \begin{align*}J: \lim_{{\longrightarrow}} \mathscr{P}_k^{\pm} \to \lim_{\longleftarrow}\mathscr{P}_k^{\pm}. \end{align*} $$

By construction, $\Pi _kJI_k: \mathscr {P}_k^{\pm }\to \mathscr {P}_k^{\pm }$ is the identity function. We denote $J_k=JI_k: \mathscr {P}_k^{\pm }\to \mathscr {P}_{\infty }^{\pm }$ .

2.6

For any $n\geq 1$ , a sequence of operators $A_k:\mathscr {P}_k^{\pm }\to \mathscr {P}_k^{\pm }$ , $k\geq n$ compatible with the inverse system induces a (limit) operator $A: \mathscr {P}_{\infty }^{\pm }\to \mathscr {P}_{\infty }^{\pm }$ . We have $A_k=\Pi _k A J_k$ . For example, the sequence $A_k=s_n$ , $k\geq n$ , given by the action of the simple transposition $s_n$ , induces a limit operator $s_n$ acting on $\mathscr {P}_{\infty }^{\pm }$ . In turn, this leads to an action of the infinite symmetric group $S(\infty )$ (the inductive limit of $S_k$ , $k\geq 1$ ) on $P_{\infty }^{\pm }$ .

For any $k\geq 0$ , denote

$$ \begin{align*} \mathscr{P}(k)^{\pm}=\{F\in \mathscr{P}_{\infty}^{-}\ |\ s_i F=F,~ \text{for all } i>k \}. \end{align*} $$

From the definition, it is clear that ${\mathscr {P}}(k)^{\pm }\subset {\mathscr {P}}(k+1)^{\pm }$ . Also, ${\mathscr {P}}(0)^{+}$ is the ring of symmetric functions $\operatorname {\mathrm {Sym}}[\mathbf {X}]$ , and, more generally, for any $k\leq 1$ , the multiplication map

$$ \begin{align*} \mathscr{P}_k^{+}\otimes \operatorname{\mathrm{Sym}}[\mathbf{X}_k] \cong\mathscr{P}(k)^+ \end{align*} $$

is an algebra isomorphism.

2.7

An important role will be played by the graded subrings $\mathscr {P}_{\mathrm {as}}^{\pm }\subset \mathscr {P}_{\infty }^{\pm }$ , defined as the inductive limit of the spaces $\mathscr {P}(k)^{\pm }$ :

$$ \begin{align*} {\mathscr{P}}_{\text{as}}^{\pm}=\bigcup_{k\geq 0}{\mathscr{P}}(k)^{\pm}. \end{align*} $$

More concretely, an element of $\mathscr {P}_{\mathrm {as}}^{\pm }$ must be fixed by all simple transpositions with the possible exception of finitely many. Following Knop [Reference Knop21], we refer to $\mathscr {P}_{\mathrm {as}}^{\pm }$ as the almost symmetric modules.

3.1 Affine Hecke Algebras

Definition 3.1. The affine Hecke algebra $\mathscr {A}_k$ of type $GL_k$ is the $\mathbb {Q}(\textbf {t})$ -algebra generated by

$$ \begin{align*} T_1,\dots,T_{k-1},X_1^{\pm1},\dots,X_k^{\pm1} \end{align*} $$

satisfying the following relations:

(3.1a) $$ \begin{align} \begin{gathered} T_{i}T_{j}=T_{j}T_{i}, \quad |i-j|>1,\\ T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}, \quad 1\leq i\leq k-2, \end{gathered} \end{align} $$

(3.1b) $$ \begin{align} (T_{i}-1)(T_{i}+\textbf{t})=0, \quad 1\leq i\leq k-1, \end{align} $$

(3.1c) $$ \begin{align} \begin{gathered} \textbf{t} T_i^{-1} X_i T_i^{-1}=X_{i+1}, \quad 1\leq i\leq k-1\\ T_{i}X_{j}=X_{j}T_{i},\quad j\neq i,i+1,\\ X_i X_j=X_j X_i \quad 1\leq i,j\leq k. \end{gathered} \end{align} $$

The definition above is based on the Bernstein presentation of the affine Hecke algebra $\mathscr {A}_k$ . The following presentation of $\mathscr {A}_k$ will also be used.

Proposition 3.2. The affine Hecke algebra $\mathscr {A}_k$ is generated by

$$ \begin{align*} \widetilde{T}_0,T_1,\dots,T_{k-1},\widetilde{\omega}_k^{\pm 1} \end{align*} $$

satisfying the relations:

(3.2a) $$ \begin{align} \begin{gathered} T_{i}T_{j}=T_{j}T_{i}, \quad |i-j|>1,\\ T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}, \quad 1\leq i\leq k-2,\\ (T_{i}-1)(T_{i}+\textbf{t})=0, \quad 1\leq i\leq k-1, \end{gathered} \end{align} $$

(3.2b) $$ \begin{align} \begin{gathered} T_{i} \widetilde{T}_0=\widetilde{T}_0 T_{i}, \quad 2\leq i\leq k-1,\\ \widetilde{T}_0 T_{1}\widetilde{T}_0=T_{1}\widetilde{T}_0 T_{1}, \\ (\widetilde{T}_0-1)(\widetilde{T}_0+\textbf{t})=0, \end{gathered} \end{align} $$

(3.2c) $$ \begin{align} \widetilde{\omega}_k T_i \widetilde{\omega}_k^{-1}=T_{i-1} \textrm{ for }1\leq i\leq k-1,\quad \widetilde{\omega}_k \widetilde{T}_0 \widetilde{\omega}_k^{-1}=T_{k-1}. \end{align} $$

Remark 3.3. $\widetilde {T}_0,\widetilde {\omega }_k$ and $X_1,\dots ,X_k$ are related by

$$ \begin{align*}\widetilde{\omega}_k=\textbf{t}^{1-k} T_{k-1}\dots T_{1}X_{1}^{-1}= X_k^{-1}T_{k-1}^{-1}\dots T_{1}^{-1}, \end{align*} $$

$$ \begin{align*} \widetilde{T}_0=\widetilde{\omega}_k^{-1}T_{k-1}\widetilde{\omega}_k=\widetilde{\omega}_k T_1\widetilde{\omega}_k^{-1} =\textbf{t}^{k-1}X_1X_k^{-1}T_1^{-1}\dots T_{k-1}^{-1}\dots T_1^{-1}. \end{align*} $$

3.2 Double Affine Hecke Algebras

Definition 3.5. The double affine Hecke algebra (DAHA) $\mathscr {H}_k$ , $k\geq 2$ , of type $GL_{k}$ is the $\mathbb {Q}(\textbf {t},\textbf {q})$ -algebra generated by $T_0,T_1,\dots ,T_{k-1}$ , $X_1^{\pm 1},\dots ,X_k^{\pm 1}$ , and $\omega _k^{\pm 1}$ satisfying equations (3.1a), (3.1b) and (3.1c) and the following relations:

(3.3a) $$ \begin{align} \begin{gathered} T_{i} T_0=T_0 T_{i}, \quad 2\leq i\leq k-1,\\ T_0 T_{1}T_0=T_{1}T_0 T_{1}, \end{gathered} \end{align} $$

(3.3b) $$ \begin{align} (T_0-1)(T_0+\textbf{t})=0, \end{align} $$

(3.3c) $$ \begin{align} \begin{gathered} \omega_k T_i \omega_k^{-1}=T_{i-1}, \quad 2\leq i\leq k-1,\\ \omega_k T_1 \omega_k^{-1}=T_0, \quad \omega_k T_0 \omega_k^{-1}=T_{k-1}, \end{gathered} \end{align} $$

(3.3d) $$ \begin{align} \begin{gathered} \omega_k X_{i+1}\omega_k^{-1}=X_i, \quad 1\leq i\leq k-1,\quad \omega_k X_1 \omega_k^{-1}=\textbf{q}^{-1}X_k. \end{gathered} \end{align} $$

The double affine Hecke algebras were first introduced, in greater generality, by Cherednik [Reference Cherednik7]. Under one of the possible sets of conventions, the definition of the DAHA of type $GL_k$ can be found, for example, in [Reference Cherednik8, §3.7]. The presentation in Definition 3.5 is consistent with the conventions in [Reference Ion and Sahi19]. We will also make use of the following equivalent presentation of $\mathscr {H}_k$ .

Proposition 3.6. The algebra $\mathscr {H}_k$ , $k\geq 2$ , is the $\mathbb {Q}(\textbf {t},\textbf {q})$ -algebra generated by the elements $T_1,\dots ,T_{k-1}$ , $X_1^{\pm 1},\dots ,X_k^{\pm 1}$ , $Y_1^{\pm 1},\dots ,Y_k^{\pm 1}$ satisfying equations (3.1a), (3.1b) and (3.1c) and the following relations:

(3.4a) $$ \begin{align} \begin{gathered} \textbf{t}^{-1} T_i Y_i T_i=Y_{i+1}, \quad 1\leq i\leq k-1\\ T_{i}Y_{j}=Y_{j}T_{i}, \quad j\neq i,i+1,\\ Y_i Y_j=Y_j Y_i \quad 1\leq i,j\leq k, \end{gathered} \end{align} $$

(3.4b) $$ \begin{align} Y_1 T_1 X_1=X_2 Y_1T_1, \end{align} $$

(3.4c) $$ \begin{align} Y_1 X_1\dots X_k= \textbf{q} X_1\dots X_k Y_1. \end{align} $$

Remark 3.7. $Y_1,\dots ,Y_k$ and $\omega _k$ are related by

$$ \begin{align*} Y_i = \textbf{t}^{k+1-i}T_{i-1} \dots T_{1}\omega_k^{-1} T_{k-1}^{-1}\dots T_{i}^{-1}. \end{align*} $$

The presentation in Proposition 3.6 can be derived from Definition 3.5. See also [Reference Schiffmann and Vasserot25, §2.1]; the generators $X_i,Y_i$ in [Reference Schiffmann and Vasserot25] correspond to $X_i^{-1}, Y_i^{-1}$ in our notation, and the relation [Reference Schiffmann and Vasserot25, (2.7)] is equivalent to equation (3.4b) modulo the relations (3.1c) and (3.4a).

3.3

The representation below is called the standard representation of $\mathscr {H}_k$ [Reference Cherednik7, Theorem 2.3]

Proposition 3.9. The following formulas define a faithful representation of $\mathscr {H}_k$ on $\mathscr {P}_k$ :

(3.5) $$ \begin{align} \begin{split} T_i f(x_1,\dots,x_k) &= s_i f(x_1,\dots,x_k) +(1-\textbf{t})x_i\frac{1-s_i}{x_i-x_{i+1}}f(x_1,\dots,x_k), \\ \widetilde{\omega}_k f(x_1,\dots,x_k) &= \textbf{t}^{1-k}T_{k-1}\dots T_{1}x_1^{-1} f(x_1,\dots,x_k),\\ \omega_k f(x_1,\dots,x_k) &= f(\textbf{q}^{-1} x_k,x_1,\dots,x_{k-1}). \end{split} \end{align} $$

Furthermore, for any $1\leq i\leq k$ , the elements $X_i$ act as left multiplication by $x_i$ .

By restriction, we obtain the corresponding standard representations of $\mathscr {H}^+_k$ and $\mathscr {H}^-_k$ .

All the standard representations are induced representations from a one-dimensional representation of the corresponding affine Hecke subalgebra.

4.1

We introduce a pair of closely related algebras with infinitely many generators, which we call stable limit DAHAs.

Definition 4.1. Let $\mathscr {H}^+$ be the $\mathbb {Q}(\textbf {t},\textbf {q})$ -algebra generated by the elements $T_i$ , $X_i$ , and $Y_i$ , $i\geq 1$ , satisfying the following relations:

(4.1a) $$ \begin{align} \begin{gathered} T_{i}T_{j}=T_{j}T_{i}, \quad |i-j|>1,\\ T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}, \quad i\geq 1, \end{gathered} \end{align} $$

(4.1b) $$ \begin{align} (T_{i}-1)(T_{i}+\textbf{t})=0, \quad i\geq 1, \end{align} $$

(4.1c) $$ \begin{align} \begin{gathered} \textbf{t} T_i^{-1} X_i T_i^{-1}=X_{i+1}, \quad i\geq 1\\ T_{i}X_{j}=X_{j}T_{i}, \quad j\neq i,i+1,\\ X_i X_j=X_j X_i,\quad i,j\geq 1, \end{gathered} \end{align} $$

(4.1d) $$ \begin{align} \begin{gathered} \textbf{t}^{-1} T_i Y_i T_i=Y_{i+1}, \quad i\geq 1\\ T_{i}Y_{j}=Y_{j}T_{i}, \quad j\neq i,i+1,\\ Y_i Y_j=Y_j Y_i, \quad i,j\geq 1, \end{gathered} \end{align} $$

(4.1e) $$ \begin{align} Y_1 T_1 X_1=X_2 Y_1T_1. \end{align} $$

We will call $\mathscr {H}^+$ the $^+$ stable limit DAHA.

Note that, as opposed to the corresponding elements of $\mathscr {H}_k$ , the elements $X_i$ , $Y_i$ are not invertible in $\mathscr {H}^+$ . Similarly, we define the $^-$ stable limit DAHA.

Definition 4.2. Let $\mathscr {H}^-$ be the $\mathbb {Q}(\textbf {t},\textbf {q})$ -algebra generated by the elements $T_i$ , $X^{-1}_i$ , and $Y^{-1}_i$ , $i\geq 1$ , satisfying the following relations:

(4.2a) $$ \begin{align} \begin{gathered} T_{i}T_{j}=T_{j}T_{i}, \quad |i-j|>1,\\ T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}, \quad i\geq 1, \end{gathered} \end{align} $$

(4.2b) $$ \begin{align} (T_{i}-1)(T_{i}+\textbf{t})=0, \quad i\geq 1, \end{align} $$

(4.2c) $$ \begin{align} \begin{gathered} \textbf{t}^{-1} T_i X_i^{-1} T_i=X_{i+1}^{-1}, \quad i\geq 1\\ T_{i}X^{-1}_{j}=X^{-1}_{j}T_{i}, \quad j\neq i,i+1,,\\ X^{-1}_i X^{-1}_j=X^{-1}_j X^{-1}_i,\quad i,j\geq 1, \end{gathered} \end{align} $$

(4.2d) $$ \begin{align} \begin{gathered} \textbf{t} T_i^{-1} Y^{-1}_i T^{-1}_i=Y^{-1}_{i+1}, \quad i\geq 1\\ T_{i}Y^{-1}_{j}=Y^{-1}_{j}T_{i}, \quad j\neq i,i+1,\\ Y^{-1}_i Y^{-1}_j=Y^{-1}_j Y^{-1}_i, \quad i,j\geq 1, \end{gathered} \end{align} $$

(4.2e) $$ \begin{align} X^{-1}_1 T^{-1}_1 Y^{-1}_1=T^{-1}_1 Y^{-1}_1 X^{-1}_2. \end{align} $$

We will call $\mathscr {H}^-$ the $^-$ stable limit DAHA.

The map

$$ \begin{align*} \mathfrak{e}: \mathscr{H}^+\to \mathscr{H}^- \end{align*} $$

that sends $T_i$ , $X_i$ , $Y_i$ to $T_i$ , $Y^{-1}_i$ , $X^{-1}_i$ , respectively, extends to an anti-isomorphism of $\mathbb {Q}(\textbf {t},\textbf {q})$ -algebras. For this reason, we regard $\mathscr {H}^+$ and $\mathscr {H}^-$ as capturing the same structure, and we will use the terminology stable limit DAHA to refer to either of them, depending on the context.

The terminology is justified by the fact that these structures arise from analyzing the stabilization phenomena for the standard representations of the algebras $\mathscr {H}^{\pm }_k$ as k approaches infinity. We will construct natural representations of both $\mathscr {H}^{\pm }$ in this fashion. To point more directly to his connection consider the following.

4.2

The stable limit DAHA is also closely related to the inductive limit of the braid groups $\mathcal {B}_k$ of k distinct points on the punctured torus. Indeed, we have the following presentation of $\mathcal {B}_k$ [Reference Bellingeri1, Theorem 1.1] (see also [Reference Mellit24, Theorem 5.1]).

Theorem 4.5. For $k\geq 2$ , the group $\mathcal {B}_k$ is generated by the elements $\sigma _i$ , $1\leq i\leq k-1$ , and $X_i$ , $Y_i$ , $1\leq i\leq k$ satisfying the following relations:

(4.3a) $$ \begin{align} \begin{gathered} \sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}, \quad |i-j|>1,\\ \sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}, \quad1\leq i\leq k-2, \end{gathered} \end{align} $$

(4.3b) $$ \begin{align} \begin{gathered} \sigma_i^{-1} X_i \sigma_i^{-1}=X_{i+1}, \quad 1\leq i\leq k-1\\ \sigma_{i}X_{j}=X_{j}\sigma_{i}, \quad j\neq i,i+1,\\ X_i X_j=X_j X_i,\quad 1\leq i,j\leq k, \end{gathered} \end{align} $$

(4.3c) $$ \begin{align} \begin{gathered} \sigma_i Y_i \sigma_i=Y_{i+1}, \quad1\leq i\leq k-1\\ \sigma_{i}Y_{j}=Y_{j}\sigma_{i}, \quad j\neq i,i+1,\\ Y_i Y_j=Y_j Y_i, \quad 1\leq i,j\leq k, \end{gathered} \end{align} $$

(4.3d) $$ \begin{align} Y_1 \sigma_1 X_1=X_2 Y_1\sigma_1. \end{align} $$

The notation here and the one in [Reference Bellingeri1] are related as follows: $X_1=b_1$ , $Y_1=a_1^{-1}$ .

Denote by $\mathcal {B}_k^+$ the submonoid of $\mathcal {B}_k$ generated by $\sigma _i,\sigma _i^{-1}$ , $1\leq i\leq k$ , and $X_i$ , $Y_i$ , $1\leq i\leq k$ . The canonical inclusion maps between the monoids $\mathcal {B}^+_k$ consitute a direct system. The direct limit monoid is denoted by $\mathcal {B}^+_{\infty }$ . A comparison between the relations in Theorem 4.5 and those in Definition 4.1 (with $\sigma _i$ mapping to $\textbf {t}^{-\frac {1}{2}}T_i$ , $i\geq 1$ ) shows that $\mathscr {H}^+$ is the quotient of the monoid ring of $\mathcal {B}^+_{\infty }$ by the ideal generated by the quadratic relations (4.1b). A similar relationship connects $\mathscr {H}^-$ and the monoid $\mathcal {B}^-_{\infty }$ , the inductive limit of the submonoids $\mathcal {B}_k^-$ of $\mathcal {B}_k$ generated by $\sigma _i,\sigma _i^{-1}$ , $1\leq i\leq k$ , and $X_i^{-1}$ , $Y_i^{-1}$ , $1\leq i\leq k$ .

4.3

Let

$$ \begin{align*} \iota: \mathscr{H}^+\to \mathscr{H}^+ \end{align*} $$

be the $\mathbb {Q}$ -algebra automorphism that sends $T_i$ to $T_i^{-1}$ , swaps $X_i$ and $Y_i$ and inverts the parameters $\textbf {t}$ and $\textbf {q}$ . It is easy to check that all relations are preserved and that $\iota $ is an involution.

Finally, let us define two $\mathbb {Q}(\textbf {t},\textbf {q})$ -algebra endomorphisms

$$ \begin{align*} \mathfrak{a},\mathfrak{b}: \mathscr{H}^+\to \mathscr{H}^+ \end{align*} $$

as follows. For all $i\geq 1$ ,

$$ \begin{align*} \mathfrak{a}(T_i)&=T_i, & \mathfrak{a}(X_i)&=X_i, &\mathfrak{a}(Y_i)&=\textbf{t}^{1-i}(T_{i-1}\cdots T_1)(T_1\cdots T_{i-1})X_iY_i, \\ \mathfrak{b}(T_i)&=T_i, & \mathfrak{b}(Y_i)&=Y_i, &\mathfrak{b}(X_i)&=\textbf{t}^{i-1}(T^{-1}_{i-1}\cdots T^{-1}_1)(T^{-1}_1\cdots T^{-1}_{i-1})Y_iX_i. \end{align*} $$

The fact that $\mathfrak {a}$ preserves the defining relations can be directly checked. The only relations that require a small computation are the commutativity relations between the $\mathfrak {a}(Y_i)$ . All the commutativity relations follow, in fact, from the commutativity relation between $\mathfrak {a}(Y_1)$ and $\mathfrak {a}(Y_2)$ . This can be verified as follows:

$$ \begin{align*} T_1X_1 Y_1T_1X_1 Y_1 &= T_1X_1 X_2Y_1T_1 Y_1 & & \text{by (4.1e)}\\ &= X_1X_2 T_1Y_1T_1 Y_1 & & \text{by (4.1c)} \\ &= X_1 X_2 Y_1 T_1Y_1T_1 & & \text{by (4.1d)} \\ &= X_1 Y_1T_1X_1 Y_1 T_1. & & \text{by (4.1e)} \end{align*} $$

The fact that $\mathfrak {a}$ is a morphism implies that $\mathfrak {b}$ is a morphism since

$$ \begin{align*}\mathfrak{b}=\iota \mathfrak{a}\iota. \end{align*} $$

Remark that the quadratic relation (4.1b) was not used in the verification of the other relations. Therefore, $\mathfrak {a}$ , $\mathfrak {b}$ and $\iota $ are not only endomorphisms of $\mathscr {H}^+$ but also endomorphisms of the underlying monoid.

Remark 4.7. We can regard the braid group on three strands $B_3$ as generated by $\mathfrak {a}, \mathfrak {b}^{-1}$ satisfying the braid relation

$$ \begin{align*}\mathfrak{a}\mathfrak{b}^{-1}\mathfrak{a}=\mathfrak{b}^{-1}\mathfrak{a}\mathfrak{b}^{-1}. \end{align*} $$

There is a surjective group morphism $B_3\to \operatorname {\mathrm {SL}}(2,\mathbb Z)$ which maps $\mathfrak {a}$ to $A= \begin {bmatrix}1&1\\ 0&1 \end {bmatrix}$ and $\mathfrak {b}$ to $ B=\begin {bmatrix}1&0\\ 1&1 \end {bmatrix}$ . The image of the free monoid $B_3^+$ generated by $\mathfrak {a}$ and $\mathfrak {b}$ maps bijectively to the free monoid generated by A and B which consists of the set $\operatorname {\mathrm {SL}}(2,\mathbb Z)^+$ of matrices with nonnegative integer entries.

Similar facts, pertaining to the algebra $\mathscr {H}^-$ , can be recorded with the help of the antimorphism $\mathfrak {e}$ .

4.4

It is important to remark that the algebras $\mathscr {H}_k^+$ , $k\geq 2$ , do not form an inverse system in any natural way and therefore $\mathscr {H}^+$ cannot be directly thought of as an inverse limit of $\mathscr {H}_k^+$ , $k\geq 2$ . However, we will show that both $\mathscr {H}^{\pm }$ acquire natural representations that are constructed by considering inverse systems built from the standard representations of $\mathscr {H}_k^{\pm }$ , $k\geq 2$ .

5.1

The results in the remainder of this section are due to Knop [Reference Knop21]. We briefly recall here the emerging structure. We refer to §2.5 for the relevant notation. The map $\pi _k:\mathscr {P}^-_k\rightarrow \mathscr {P}^-_{k-1}$ is partially compatible with the actions of $\mathscr {A}_k^-$ and $\mathscr {A}_{k-1}^-$ . More precisely,

(5.1) $$ \begin{align} \begin{split} \pi_k T_i &= T_i \pi_k,\quad 1\leq i\leq k-2,\\ \pi_k \widetilde{\omega}_k &= 0, \\ \pi_k T_{k-1}\widetilde{\omega}_k & = \widetilde{\omega}_{k-1}\pi_k,\\ \pi_k X_i^{-1} &= X_{i}^{-1}\pi_k,\quad 1\leq i\leq k-1,\\ \pi_k X_k^{-1} &= 0. \end{split} \end{align} $$

We refer to [Reference Knop21, Theorem 9.1] for the details.

5.2

The compatibility with the actions of $\mathscr {H}_k^{-}$ and $\mathscr {H}_{k-1}^{-}$ can also be investigated, but the verification is more delicate. More precisely, we have the following result [Reference Knop21, Proposition 9.11].

Proposition 5.1. For any $1\leq i\leq k-1$ , we have

(5.2) $$ \begin{align} \pi_k Y_i= Y_{i}\pi_k. \end{align} $$

Furthermore, the operator $Y^{-1}_i$ stabilizes both $\mathscr {P}_k^{-}$ and $\mathscr {P}_{k-1}^-$ and

(5.3) $$ \begin{align} \pi_k Y^{-1}_i= Y^{-1}_{i}\pi_k. \end{align} $$

5.3

For any $n\geq 1$ , the sequence of operators $(A_k)_{k\geq 1}$ defined by

$$ \begin{align*} A_k:=Y_n^{(k)}, \quad k\geq n, \end{align*} $$

induces the limit operator $\displaystyle \mathscr {Y}_n:\lim _{\substack {\longleftarrow \\ k\geq n}} \mathscr {P}_k^-\to \lim _{\substack {\longleftarrow \\ k\geq n}} \mathscr {P}_k^-$ . Since $\displaystyle \lim _{\substack {\longleftarrow \\ k\geq n}} \mathscr {P}_k$ and $\mathscr {P}^-_{\infty }$ are canonically isomorphic, we obtain an operator $\displaystyle \mathscr {Y}_n:\mathscr {P}^-\to \mathscr {P}^-$ . Similarly, we obtain the limit operators $\mathscr {T}_i, \mathscr {X}^{-1}_i$ , and $\mathscr {Y}^{-1}_i$ , $i\geq 1$ . It is important to remark that the operators $\mathscr {Y}^{-1}_i$ are invertible (with inverse $\mathscr {Y}_i$ ). The following result immediately follows.

5.4

As it was pointed out in [Reference Knop21, §10] the almost symmetric module $\mathscr {P}_{\mathrm {as}}^-$ , defined in §2.7, is a $\mathscr {H}^-$ -stable subspace of $\mathscr {P}_{\infty }^-$ and is more canonical from a certain point of view. Specifically, each $\mathscr {P}_k^-$ is a parabolic module for the affine Hecke algebra $\mathscr {A}_k^-$ and has a standard basis (in the sense of Kazhdan-Lusztig theory) indexed by compositions with at most k parts. The sequences consisting of the standard basis elements indexed by the same composition (in all $\mathscr {P}_k^-$ , $k\geq n$ , for some n) give elements of $\mathscr {P}_{\infty }^-$ (see [Reference Knop21, §9]) which are expected to play the role of a standard basis for the limit representation. However, these limits of standard basis elements do not span $\mathscr {P}_{\infty }^-$ but rather the smaller space $\mathscr {P}_{\mathrm {as}}^-$ .

We call this representation the standard representation of $\mathscr {H}^-$ . We expect this representation to be faithful.

As explained in [Reference Knop21], a sequence on nonsymmetric Macdonald polynomials indexed by the same composition gives rise to an element of $\mathscr {P}_{\mathrm {as}}^-$ , and such elements are common eigenfunctions for the action the operators $\mathscr {Y}^{-1}_i$ . The limit nonsymmetric Macdonald polynomials do not span $\mathscr {P}_{\mathrm {as}}^-$ , and therefore the spectral theory of the operators $\mathscr {Y}^{-1}_i$ acting on $\mathscr {P}_{\mathrm {as}}^-$ is not yet fully understood.

6.1

Before delving into this section, which is the technical core of the article, it might be helpful to offer a summary of the difficulties that one encounters in the effort to understand the limiting behaviour of the Cherednik operators (the analysis of all the other generators of $\mathscr {H}_k^+$ is straightforward) in relationship to the inverse system $(\mathscr {P}_k^+)_{k\geq 2}$ , as well as provide a less technical description of the new elements that are employed to describe this limiting behaviour.

It has been known to the specialists that the Cherednik operators $Y^{(k)}_i$ are not compatible with the inverse system $(\mathscr {P}_k^+)_{k\geq 2}$ . Our first observation is that the family of operators $\textbf {t}^kY^{(k)}_i$ , for fixed i, are compatible with inverse system $(x_i\mathscr {P}_k^+)_{k\geq 2}$ , which does lead to a limit operator acting on $x_i\mathscr {P}_{\infty }^+$ . However, there is no common domain for all limit operators $Y_i$ , $i\geq 1$ . To address this situation, we consider some modified operators acting on $\mathscr {P}_k^+$ , which we denote by $\widetilde {Y}_i^{(k)}$ . Like the Cherednik operators, these operators satisfy the relations $\widetilde {Y}^{(k)}_{i+1}=\textbf {t}^{-1} T_i \widetilde {Y}^{(k)}_i T_i$ , $1\leq i\leq k-1$ , $T_j \widetilde {Y}^{(k)}_i=\widetilde {Y}^{(k)}_i T_j$ , $|i-j|>1$ , and are therefore fully determined by the operator $\widetilde {Y}^{(k)}_{1}$ , which has the following additional properties

1. i) $\widetilde {Y}_1^{(k)}=\textbf {t}^k Y_1^{(k)}$ on $x_1\mathscr {P}_k^+$ ;

2. ii) $\widetilde {Y}_1^{(k)}\cdot \mathscr {P}_k^+\subseteq x_1\mathscr {P}_k^+$ .

The first property implies that $\widetilde {Y}_i^{(k)}=\textbf {t}^k Y_i^{(k)}$ on $x_i\mathscr {P}_k^+$ for all $1\leq i \leq k$ ; the second property implies that $\widetilde {Y}_i^{(k)}\cdot \mathscr {P}_k^+\subseteq T_{i-1}\cdots T_1 x_1\mathscr {P}_k^+$ for all $1\leq i \leq k$ . The operator for $\widetilde {Y}_1^{(k)}$ is obtained by projecting the action of $Y_1^{(k)}$ onto the space $x_1\mathscr {P}_k^+$ .

The structural properties of the algebra generated by $\widetilde {Y}_i^{(k)}$ , $X_i$ , $1\leq k$ , and $T_i$ , $1\leq i\leq k-1$ are used to define an algebraic structure $\widetilde {\mathscr {H}}_k^+$ called the deformed DAHA. $\widetilde {\mathscr {H}}_k^+$ has a natural action $\mathscr {P}_k^+$ (the standard representation), which is related to the standard representation of $\mathscr {H}_k^+$ on $\mathscr {P}_k^+$ in the manner described above.

The modified Cherednik operators $\widetilde {Y}_i^{(k)}$ , for fixed i, are not yet compatible with the inverse system $(\mathscr {P}_k^+)_{k\geq 2}$ , but they are quite close to satisfying such a property. To be able to point out more precisely what happens, we first remark that they act consistently on constant sequences (i.e., sequences compatible with the canonical inclusions $\iota _k: \mathscr {P}_{k-1}^+\to \mathscr {P}_k^+$ ) in this inverse system (Lemma 6.10). This fact implies the existence of limit operators defined on

$$ \begin{align*} \displaystyle{\lim_{{\longrightarrow}}} \mathscr{P}_k^+=\mathbb Q(\textbf{t},\textbf{q})[x_1,x_2,\dots], \end{align*} $$

the ring of polynomials in infinitely many variables, but still not on $\mathscr {P}_{\infty }^+$ . Interestingly, the image of $\displaystyle {\lim _{{\longrightarrow }}} \mathscr {P}_k^+$ under the limit operators lies in $\mathscr {P}_{\mathrm {as}}^+$ (which is strictly smaller than $\mathscr {P}_{\infty }^+$ ), showing that the smallest space on which one can hope to acquire an action of an algebra of limit operators is $\mathscr {P}_{\mathrm {as}}^+$ .

In order to understand the failure of the diagram

to be commutative, we can consider the difference $\pi _k \widetilde {Y}_{1}^{(k)} - \widetilde {Y}_{1}^{(k-1)} \pi _k: \mathscr {P}_k^+\to \mathscr {P}_{k-1}^+$ . On the subspace $x_k\mathscr {P}_k^+$ , this difference has no kernel and reduces to $\pi _k \widetilde {Y}_{1}^{(k)} : x_k\mathscr {P}_k^+\to \mathscr {P}_{k-1}^+$ . Since for every monomial in $m\in \mathbb Q(\textbf {t},\textbf {q})[x_1,x_2,\dots ]$ there exists a unique k such that $m\in x_k\mathscr {P}_k^+$ , we can use the action of $\pi _k \widetilde {Y}_{1}^{(k)}$ on $x_k\mathscr {P}_k^+$ to define an operator

$$ \begin{align*} W_1: \mathbb Q(\textbf{t},\textbf{q})[x_1,x_2,\dots]\to \mathbb Q(\textbf{t},\textbf{q})[x_1,x_2,\dots] \end{align*} $$

such that $W_1=\pi _k \widetilde {Y}_{1}^{(k)}$ on $x_k\mathscr {P}_k^+$ . By restricting $W_1$ to $\mathscr {P}_k^+$ , we obtain the operator $W_1^{(k)}: \mathscr {P}_k^+\to \mathscr {P}_k^+$ . We can regard the operator $W_1$ as collecting some obvious obstructions to the commutativity of the above diagram.

As it turns out, these are all the obstructions to the commutativity of the diagram: If $ \widetilde {Z}_{1}^{(k)}= \widetilde {Y}_{1}^{(k)} - W_1^{(k)}$ , then the corresponding diagram for the $ \widetilde {Z}_{1}^{(k)} $ operators is commutative, leading to an operator

$$ \begin{align*} \widetilde{Z}_{1}^{(\infty)}: \mathscr{P}_{\infty}^+\to \mathscr{P}_{\infty}^+. \end{align*} $$

Furthermore, the action of $W_1$ (or $W_1^{(k)}$ ) on monomials can be explicitly computed; this is the action described in §6.8. Since this action is particularly simple, we were able to compute the action of the operators $W_1^{(k)}$ on sequences in the inverse system $(\mathscr {P}_k^+)_{k\geq 2}$ . This computation revealed that if the sequence in the inverse system has limit in $\mathscr {P}_{\mathrm {as}}^+$ , then one obtains by applying $W_1^{(k)}$ another sequence in $(\mathscr {P}_k^+)_{k\geq 2}$ which can be written as a finite linear expression of sequences compatible with the inverse system. The coefficients in these linear expressions are sequences in $\mathbb Q(\textbf {t},\textbf {q})$ that are convergent in the $\textbf {t}$ -adic topology (i.e., $\textbf {t}$ is treated as a small positive real number). This led us to the concept of limit described in §6.9. We refer to the sequences that have limit in this sense as convergent sequences.

Since both ${W}_i^{(k)}$ and $\widetilde {Z}_i^{(k)}$ have limit in this sense, we obtain that the operators $\widetilde {Y}_i^{(k)}$ have limit (denoted by $\mathscr {Y}_i$ ). Although, the co-domain for the limit operators ${W}_i^{(\infty )}$ and $\widetilde {Z}_i^{(\infty )}$ is, in general, $\mathscr {P}_{\infty }^+$ , it turns out that $\mathscr {Y}_i: \mathscr {P}_{\mathrm {as}}^+\to \mathscr {P}_{\mathrm {as}}^+$ . Furthermore, $\mathscr {Y}_i$ satisfies a continuity property: It sends convergent sequences to convergent sequences.

The analysis of the algebraic structure generated by the action of the limit operators $\mathscr {Y}_i$ makes use of the continuity property. Their most remarkable property of the limit operators is their commutativity. The commutativity holds, of course, for the finite rank Cherednik operators, but it does not hold for the modified Cherednik operators, and it was interesting to discover that the commutativity is restored in the limit. Furthermore, the action of $\mathscr {Y}_i$ on $x_i\mathscr {P}_{\mathrm {as}}^+$ matches the action of the corresponding limit Cherednik operator $Y_i$ . Ultimately, the action of the limit operators $\mathscr {Y}_i$ , $\mathscr {X}_i$ , and $\mathscr {T}_i$ define an action of $\mathscr {H}^+$ on $\mathscr {P}_{\mathrm {as}}^+$ .

6.2

The map $\pi _k:\mathscr {P}^+_k\rightarrow \mathscr {P}^+_{k-1}$ is also partially compatible with the actions of $\mathscr {A}_k^+$ and $\mathscr {A}_{k-1}^+$ . More precisely,

(6.1) $$ \begin{align} \pi_k T_i &= T_i \pi_k,\quad 1\leq i\leq k-2,\nonumber\\ \pi_k T_{k-1}^{-1}\dots T_1^{-1}\widetilde{\omega}_k^{-1} &= 0\nonumber \\ \pi_k \widetilde{\omega}_k^{-1}T_{k-1} & = \widetilde{\omega}_{k-1}^{-1}\pi_k\\ \pi_k X_i &= X_{i}\pi_k,\quad 1\leq i\leq k-1,\nonumber \\ \pi_k X_i &= 0.\nonumber\end{align} $$

As before, these relations show that the operators $T_i$ and $X_i$ have limits (in the sense of §5.3), denoted by $\mathscr {T}_i$ and $\mathscr {X}_i$ , that act on $\mathscr {P}_{\infty }^+$ .

6.3

On the other hand, for any $i\geq 1$ , the actions of the operators $Y^{(k)}_{i}$ and $Y^{(k-1)}_{i}$ are no longer compatible with the map $\pi _k$ . One immediate obstruction is the fact that the eigenvalues of $Y_{i}$ on $\mathscr {P}_k^+$ and $\mathscr {P}_{k-1}^+$ no longer match, as is was the case for their action on $\mathscr {P}_k^-$ and $\mathscr {P}_{k-1}^-$ . The common spectrum of the action of $Y_i$ , $1\leq i\leq k$ , on $\mathscr {P}_k^+$ is known from type $A_{k-1}$ Macdonald theory. The common eigenfunctions are nonsymmetric Macdonald polynomials; they are indexed by compositions $\lambda =(\lambda _1,\dots ,\lambda _k)\in \mathbb Z_{\geq 0}^k$ . The eigenvalue of $Y^{(k)}_i$ corresponding to the nonsymmetric Macdonald polynomial indexed by $\lambda $ is

$$ \begin{align*} e_{\lambda}(i):=\textbf{q}^{\lambda_i}\textbf{t}^{1-w_{\lambda}(i)}, \end{align*} $$

where $w_{\lambda }(i)=|\{j=1,\dots ,i~|~\lambda _j\leq \lambda _i\}|+|\{j=1,\dots ,i~|~\lambda _j< \lambda _i\}|$ . For the composition $(\lambda ,0)\in \mathbb Z_{\geq 0}^{k+1}$ , we have

$$ \begin{align*} e_{(\lambda,0)}(i)=e_{\lambda}(i)\textbf{t}^{-1},\quad \text{if } \lambda_i>0, \quad \text{and}\quad e_{(\lambda,0)}(i)=e_{\lambda}(i),\quad \text{if } \lambda_i=0. \end{align*} $$

One way to partially correct this problem is to consider the normalized operators $\textbf {t}^{k}Y^{(k)}_i$ . Comparing the eigenvalues corresponding to $\lambda $ and $(\lambda ,0)$ gives

$$ \begin{align*} \textbf{t}^{k+1}e_{(\lambda,0)}(i)=\textbf{t}^k e_{\lambda}(i),\quad \text{if } \lambda_i>0, \quad \text{and}\quad \textbf{t}^{k+1} e_{(\lambda,0)}(i)=\textbf{t}^k e_{\lambda}(i)\cdot \textbf{t},\quad \text{if } \lambda_i=0. \end{align*} $$

Example 6.1. The discrepancy between the eigenvalues is also reflected in the eigenfunctions. For example, the Macdonald polynomial corresponding to the compositions $(0,1,0)$ and $(0,1)$ are, respectively,

$$ \begin{align*} x_2+\frac{\textbf{q}(1-\textbf{t})}{\textbf{q}-\textbf{t}^2}x_1\quad\quad \text{and}\quad\quad x_2+\frac{\textbf{q}(1-\textbf{t})}{\textbf{q}-\textbf{t}}x_1. \end{align*} $$

This shows, in particular, that $\pi _3 Y_1^{(3)}\neq Y_1^{(2)}\pi _3$ .

6.4

A crucial observation is that the actions of the operators $\textbf {t}^{k}Y^{(k)}_iX^{(k)}_i\in \mathscr {H}_k^{+}$ on $\mathscr {P}_{k}^{+}$ are compatible with the inverse system.

Proposition 6.2. For any $1\leq i\leq k-1$ , we have

$$ \begin{align*} \pi_k\textbf{t}^{k}Y_i X_i=\textbf{t}^{k-1}Y_iX_i\pi_k. \end{align*} $$

Proof. First note that we have

(6.2) $$ \begin{align} \pi_k\omega^{-1}_k T_{k-1}= \omega^{-1}_{k-1}\pi_k, \end{align} $$

which can be verified by direct computation. Hence, by equation 6.1 we have

$$ \begin{align*} \pi_k Y_i^{(k)}X_i^{(k)}&= \textbf{t}^{1-i}T_{i-1} \dots T_{1}\pi_k(\omega^{-1}_k T_{k-1}^{-1}\dots T_{i}^{-1}X_i)\\ &=\textbf{t}^{1-k}T_{i-1} \dots T_{1}\pi_k(\omega^{-1}_k X_kT_{k-1}\dots T_{i})\\ &=\textbf{t}^{1-k}T_{i-1} \dots T_{1}\omega^{-1}_{k-1}\pi_k( T_{k-1}^{-1}X_kT_{k-1}\dots T_{i})\\ &=\textbf{t}^{-k}T_{i-1} \dots T_{1}\omega^{-1}_{k-1}\pi_k( T_{k-1}X_kT_{k-1}\dots T_{i})\\ &=\textbf{t}^{1-k}T_{i-1} \dots T_{1}\omega^{-1}_{k-1}\pi_k( X_{k-1}T_{k-2}\dots T_{i})\\ &=\textbf{t}^{-i}T_{i-1} \dots T_{1}\omega^{-1}_{k-1}T_{k-2}^{-1}\dots T_{i}^{-1}X_i\pi_k\\ &=\textbf{t}^{-1}Y_i^{(k-1)}X_i^{(k-1)}\pi_k \end{align*} $$

Therefore, we have $\pi _k\textbf {t}^{k}Y_i^{(k)}X^{(k)}_i=\textbf {t}^{k-1}Y_i^{(k-1)}X^{(k-1)}_i\pi _k$ .

In other words, one can obtain a limit operator corresponding to $Y_i$ , acting on the space $x_i\mathscr {P}_{\infty }^+\subset \mathscr {P}_{\infty }^+$ . However, this is not satisfactory because it does not produce a nontrivial common domain for all the limit operators $Y_i$ , $i\geq 1$ . In order to extend the action of the limit operator $Y_i$ to a larger domain, we introduce the deformed double affine Hecke algebras and the concept of limit detailed in §6.9.

6.5 The deformed double affine Hecke algebras

Definition 6.3. The deformed DAHA $\widetilde {\mathscr {H}}_{k}^{+}$ , $k\geq 2$ , is the $\mathbb {Q}(\textbf {t},\textbf {q})$ -algebra generated by the elements $T_1,\dots ,T_{k-1}$ , $X_1,\dots ,X_k$ and $\varpi _k$ satisfying equations (3.1a), (3.1b) and (3.1c) and the following relations:

(6.3) $$ \begin{align} \begin{gathered} \varpi_kT_i=T_{i+1}\varpi_k, \quad 1\leq i\leq k-2,\\ \varpi_kX_i=X_{i+1}\varpi_k,\quad 1\leq i\leq k-1, \end{gathered} \end{align} $$

(6.4) $$ \begin{align} \begin{gathered} \gamma_k T_{k-1}=-\textbf{t} \gamma_k,\quad T_1\gamma_k =\gamma_k,\\ \gamma_k\varpi_k^{k-2}\gamma_k=\gamma_k\varpi_k^{k-1}\gamma_k=\gamma_k\varpi_k^{k}=0, \end{gathered} \end{align} $$

where

$$ \begin{align*} \gamma_k=\varpi_k^2 T_{k-1}-T_{1}\varpi_k^2. \end{align*} $$

Notation 6.5. Let $\widetilde {Y}_i$ , $1\leq i\leq k$ , be defined as follows

$$ \begin{align*} \widetilde{Y}_1=\textbf{t}^{k}\varpi_kT_{k-1}^{-1}\dots T_1^{-1},\qquad \widetilde{Y}_{i+1}=\textbf{t}^{-1} T_i \widetilde{Y}_i T_i, \quad 1\leq i\leq k-1. \end{align*} $$

Therefore,

$$ \begin{align*} \widetilde{Y}_i = \textbf{t}^{1-i+k}T_{i-1} \dots T_{1}\varpi_k T_{k-1}^{-1}\dots T_{i}^{-1}, \quad 1\leq i\leq k. \end{align*} $$

Remark 6.6. It is important to remark that the relation (3.4b) is also satisfied in $\widetilde {\mathscr {H}}_k$ . Indeed

$$ \begin{align*} \widetilde{Y}_{1} T_1 X_1 &=\textbf{t}^{k}\varpi_kT_{k-1}^{-1}\ldots T_2^{-1}X_1 \\ &=\textbf{t}^{k}X_2\varpi_kT_{k-1}^{-1}\ldots T_2^{-1} \\ &=X_2 \widetilde{Y}_{1} T_1. \end{align*} $$

The relations

$$ \begin{align*}T_j \widetilde{Y}_i=\widetilde{Y}_i T_j, \quad |i-j|>1 \end{align*} $$

are also satisfied. The verification is a simple consequence of the braid relations and equation (6.3).

Remark 6.7. On the other hand, the elements $\widetilde {Y}_i$ , $1\leq i\leq k$ , do not commute. In fact, we have

$$ \begin{align*} [\widetilde{Y}_1, \widetilde{Y}_2]=\textbf{t}^{2k-1}\gamma_k T_{k-1}^{-1}\dots T_1^{-1}T_{k-1}^{-1}\dots T_2^{-1}=\textbf{t}^{2k-1}\gamma_k T_{k-2}^{-1}\dots T_1^{-1}T_{k-1}^{-1}\dots T_1^{-1}. \end{align*} $$

Therefore, $\gamma _k$ can be seen as an obstruction to the commutativity of the elements $\widetilde {Y}_i$ .

Since the notation for the generators of $\widetilde {\mathscr {H}}_k^+$ and the elements $\widetilde {Y}_i$ does not specify the integer k, we will use the notation in the fashion indicated in Convention 3.11.

6.6

For any $1\leq i \leq k$ , let $\operatorname {\mathrm {pr}}_i:\mathscr {P}_k^+ \to \mathscr {P}_k^+$ , be the $\mathbb {Q}(\textbf {t},\textbf {q})$ -linear map which acts as identity on monomials divisible by $x_i$ and as the zero map on monomials not divisible by $x_i$ . In other words, $\operatorname {\mathrm {pr}}_i$ is the projection onto the subspace $x_i\mathscr {P}_k^+$ .

We define the following action of $\widetilde {\mathscr {H}}_{k}^{+}$ on $\mathscr {P}_{k}^{+}$ . The action is related to the action of $\mathscr {H}_k$ on $\mathscr {P}_k^+$ .

Theorem 6.8. The following formulas define an action of $\widetilde {\mathscr {H}}_{k}^{+}$ on $\mathscr {P}_k^+$ :

(6.5) $$ \begin{align} \begin{split} T_i f(x_1,\dots,x_k) &= s_i f(x_1,\dots,x_k) +(1-\textbf{t})x_i\frac{1-s_i}{x_i-x_{i+1}}f(x_1,\dots,x_k), \\ X_i f(x_1,\dots,x_k) &=x_i f(x_1,\dots,x_k),\\ \varpi_k f(x_1,\dots,x_k) &=\operatorname{\mathrm{pr}}_1f(x_2,\dots,x_k, \textbf{q} x_1). \end{split} \end{align} $$

Note that the action of $ \varpi _k $ coincides with the action of $\operatorname {\mathrm {pr}}_1\omega _k^{-1}$ .

Proof. We only need to check the relations (6.3) and (6.4). From equation (3.3c), we have

$$ \begin{align*} \varpi_kT_i f&=\operatorname{\mathrm{pr}}_1\omega_k^{-1}T_i f\\ &=\operatorname{\mathrm{pr}}_1(T_{i+1}\omega_k^{-1} f)\\ &=T_{i+1}\operatorname{\mathrm{pr}}_1\omega_k^{-1} f\\ &=T_{i+1}\varpi_k f \end{align*} $$

for $f\in \mathscr {P}_{k}^{+}$ and $1\leq i\leq k-2$ . Similarly, from equation (3.3d), we have

$$ \begin{align*} \varpi_k X_i f=X_{i+1}\varpi_k f \end{align*} $$

for $f\in \mathscr {P}_{k}^{+}$ and $1\leq i\leq k-1$ .

By linearity it suffices to verify the relation (6.4) on monomials. A straightforward computation leads to the following explicit expression of the action of $\gamma _k$

(6.6) $$ \begin{align} \gamma_k(x_1^{t_1}\dots x_{k-1}^{t_{k-1}}x_{k}^{t_{k}}) = \begin{cases} 0, & \textrm{ if } t_{k-1}\neq 0,t_k\neq 0\\\\ (1-\textbf{t})\textbf{q}^{t_{k-1}}(x_1^{t_{k-1}-1}x_2+x_1^{t_{k-1}-2}x_2^2+\dots \\+x_1 x_2^{t_{k-1}-1})(x_3^{t_1}x_4^{t_2}\dots x_k^{t_{k-2}}), & \textrm{ if } t_{k-1}\neq 0,t_k= 0\\\\ (\textbf{t}-1)\textbf{q}^{t_{k}}(x_1^{t_{k}-1}x_2+x_1^{t_{k}-2}x_2^2+\dots\\+x_1 x_2^{t_{k}-1})(x_3^{t_1}x_4^{t_2}\dots x_k^{t_{k-2}}), &\textrm{ if } t_{k-1}= 0,t_k\neq 0. \end{cases} \end{align} $$

Note that the expression immediately implies that the actions of $\gamma _k T_{k-1}$ and $-\textbf {t} \gamma _k$ coincide and similarly for the actions of $T_1\gamma _k$ and $\gamma _k$ . The remaining relations can also be directly verified.

6.7

We are now ready to explore the compatibility between the action of the operators $\widetilde {Y}_i^{(k)}$ and the map $\pi _k: \mathscr {P}_k^+\to \mathscr {P}_{k-1}^+$ .

Lemma 6.10. The following diagram is commutative:

Equivalently, we have

$$ \begin{align*} \pi_{k}\widetilde{Y}_i^{(k)} f=\widetilde{Y}_i^{(k-1)} f \end{align*} $$

for all $f\in \mathscr {P}_{k-1}^{+}$ .

Proof. Note that $\operatorname {\mathrm {pr}}_1\omega ^{-1}_{k}=\omega ^{-1}_{k}\operatorname {\mathrm {pr}}_k$ . For any $f\in \mathscr {P}_{k-1}^{+}$ , we have

$$ \begin{align*} \pi_{k}\widetilde{Y}_i^{(k)}\iota_k f&=\pi_{k}\textbf{t}^{1-i+k}T_{i-1} \dots T_{1} \operatorname{\mathrm{pr}}_1\omega^{-1}_{k}T_{k-1}^{-1}\dots T_i^{-1}f\\ &=\textbf{t}^{k-i}T_{i-1} \dots T_{1}\operatorname{\mathrm{pr}}_1\pi_{k}\omega^{-1}_{k}T_{k-1}T_{k-2}^{-1}\dots T_i^{-1}f\\&\quad+\pi_{k}\textbf{t}^{k-i}(\textbf{t}-1)T_{i-1} \dots T_{1}\operatorname{\mathrm{pr}}_1\omega^{-1}_{k}T_{k-2}^{-1}\dots T_i^{-1}f\\ &=\textbf{t}^{k-i}T_{i-1} \dots T_{1}\operatorname{\mathrm{pr}}_1\omega^{-1}_{k-1}\pi_{k}T_{k-2}^{-1}\dots T_i^{-1}f\\&\quad+\pi_{k}\textbf{t}^{k-i}(\textbf{t}-1)T_{i-1} \dots T_{1}\omega^{-1}_{k}\operatorname{\mathrm{pr}}_k(T_{k-2}^{-1}\dots T_i^{-1}f)\\ &=\widetilde{Y}_i^{(k-1)}\pi_{k}f+0. \end{align*} $$

For the third equality, we used equation (6.2).

As an immediate consequence, we obtain the following.

Proposition 6.11. For any $i\geq 1$ , the sequence of operators $(\widetilde {Y}_i^{(k)})_{k\geq 2}$ induces a map

$$ \begin{align*} \widetilde{Y}_i^{(\infty)}: \lim_{{\longrightarrow}} \mathscr{P}_k^+\to \mathscr{P}_{\infty}^{+} \end{align*} $$

such that $\Pi _k \widetilde {Y}_i^{(\infty )} I_k=\widetilde {Y}_i^{(k)}$ for all $k\geq i$ , $k\geq 2$ .

Proof. Fix $n\geq i$ , and denote by $\iota _{n,k}: \mathscr {P}_n^+\to \mathscr {P}_k^+$ , $n\leq k$ , the canonical inclusion. By Lemma 6.10, the sequence of maps $\widetilde {Y}_i^{(k)} \iota _{n,k}: \mathscr {P}_n^+\to \mathscr {P}_k^+$ , $k\geq n,i$ is compatible with the structure maps $\pi _k$ . Therefore, they induce a morphism

$$ \begin{align*} \mathscr{P}_n^+\to \lim_{\substack{\longrightarrow \\ k\geq n}} \mathscr{P}_k^+\cong \displaystyle \lim_{{\longrightarrow }} \mathscr{P}_k^+. \end{align*} $$

Furthermore, Lemma 6.10 implies that these maps are compatible with the structure maps $\iota _n$ and therefore induce the desired morphism.

Example 6.12. To illustrate the construction of the limit operator $\widetilde {Y}_1^{(\infty )}: \displaystyle {\lim _{{\longrightarrow }}} \mathscr {P}_k^+\to \mathscr {P}_{\infty }^{+}$ , let us compute its action on the element $x_2^2\in \displaystyle {\lim _{{\longrightarrow }}} \mathscr {P}_k^+$ . A direct computation gives

$$ \begin{align*} \widetilde{Y}_1^{(2)}\cdot x_2^2&=\textbf{q}^2 \textbf{t}(\textbf{t}-1)x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2\\ \widetilde{Y}_1^{(3)}\cdot x_2^2&=\textbf{q}^2 \textbf{t}(\textbf{t}-1)x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 x_1x_3\\ \widetilde{Y}_1^{(4)}\cdot x_2^2&=\textbf{q}^2 \textbf{t}(\textbf{t}-1)x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 (x_1x_3+x_1x_4) \end{align*} $$

and, more generally, for $k\geq 4$ ,

$$ \begin{align*} \widetilde{Y}_1^{(k)}\cdot x_2^2=\textbf{q}^2 \textbf{t}(\textbf{t}-1)x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 (x_1x_3+\cdots x_1x_k). \end{align*} $$

Therefore, the limit operator acts as

$$ \begin{align*} \widetilde{Y}_1^{(\infty)}\cdot x_2^2=\textbf{q}^2 \textbf{t}(\textbf{t}-1)x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 x_1 e_1[\mathbf{X}_2]. \end{align*} $$

On the other hand, the action same operator on a nonconstant sequence (e.g., one that converges to a nontrivial almost symmetric function) shows a slightly different behaviour.

Example 6.13. Let $F[\mathbf {X}]=x_2^2+x_3^2+\cdots =p_2[\mathbf {X}_1]$ , which is the inverse limit of the sequence

$$ \begin{align*} F_{k}=\Pi_k F[\mathbf{X}]=x_2^2+\cdots+x_k^2,~ k\geq 2. \end{align*} $$

We have

$$ \begin{align*} \widetilde{Y}_1^{(2)}\cdot F_{2}&=\textbf{q}^2(\textbf{t}^2-\textbf{t})x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2\\ \widetilde{Y}_1^{(3)}\cdot F_{3}&=\textbf{q}^2(\textbf{t}^3-\textbf{t})x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)(x_1 x_2+x_1 x_3) \end{align*} $$

and, more generally, for $k\geq 2$ ,

$$ \begin{align*} \widetilde{Y}_1^{(k)}\cdot F_{k}=\textbf{q}^2(\textbf{t}^k-\textbf{t})x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)(x_1 x_2+x_1 x_3+\cdots+x_1x_k). \end{align*} $$

As it can be clearly seen, the sequence $ \widetilde {Y}_1^{(k)}F_{k}$ fails to have an inverse limit because of the contribution of the term $\textbf {q}^2(\textbf {t}^k-\textbf {t})x_1^2$ .

As we will see next, the general behaviour of the action of fixed operator on a sequence with inverse limit is no more complicated than the one exhibited in Example 6.13. The operators $W_i^{(k)}$ introduced in the next section formalize this observation.

6.8

We define the $\mathbb {Q}(\textbf {t},\textbf {q})$ -linear operators $\displaystyle W^{(k)}_i: \mathscr {P}_k^+\to \mathscr {P}_{k}^{+}$ , $i\geq 1$ , as follows. Let $s\geq 0$ , and let $i_1<i_2<\dots <i_s\leq k$ and $t_1, \dots , t_s$ be positive integers. We set

(6.7) $$ \begin{align} W^{(k)}_1(x_{i_1}^{t_{1}}\dots x_{i_s}^{t_{s}})=\begin{cases} 0, & \text{ if } 1=i_1 \text{ or }s=0,\\\\ \ \textbf{t}^{i_s}(1-\textbf{t}^{-1})\textbf{q}^{t_{s}}x_{1}^{t_{s}}x_{i_1}^{t_{1}}\dots x_{i_{s-1}}^{t_{s-1}}, & \text{ if } i_1>1. \end{cases} \end{align} $$

Define

$$ \begin{align*} W^{(k)}_{i+1}=\textbf{t}^{-1} T_i W^{(k)}_i T_i,\quad 1\leq i\leq k-1. \end{align*} $$

For $1\leq i\leq k$ , we also denote

$$ \begin{align*} \widetilde{Z}_{i}^{(k)}=\widetilde{Y}_{i}^{(k)}-W_{i}^{(k)}. \end{align*} $$

Lemma 6.14. The following diagram is commutative:

Proof. It is enough to prove the statement for $i=1$ . For any monomial $f\in \mathscr {P}_{k-1}^{+}$ , we have

$$ \begin{align*} \pi_k W_1^{(k)} f =W_1^{(k-1)}\pi_k f \end{align*} $$

and the commutativity of the diagram follows from Lemma 6.10.

Let $f\in x_k \mathscr {P}_{k}^{+}$ be a monomial and $g\in \mathscr {P}_{k}^{+}$ . A direct check shows that

$$ \begin{align*} \pi_k\omega^{-1}_{k}\operatorname{\mathrm{pr}}_kT_{k-1}^{-1}\operatorname{\mathrm{pr}}_k g \neq 0 \end{align*} $$

only if g is not divisible by $x_{k-1}$ . On the other hand,

$$ \begin{align*} T_i^{-1} x_i\mathscr{P}_k^+ \subseteq x_{i+1}\mathscr{P}_k^+. \end{align*} $$

Now,

$$ \begin{align*} \pi_k \widetilde{Y}_{1}^{(k)}f&= \textbf{t}^k \pi_k\varpi_k T_{k-1}^{-1}\dots T_{1}^{-1} f\\ &=\textbf{t}^k \pi_k\varpi_k T_{k-1}^{-1}\dots T_{1}^{-1}\operatorname{\mathrm{pr}}_k f\\ &=\textbf{t}^k \pi_k\omega^{-1}_{k}\operatorname{\mathrm{pr}}_kT_{k-1}^{-1}\operatorname{\mathrm{pr}}_k (T_{k-2}^{-1}\dots T_{1}^{-1} f). \end{align*} $$

Based on the previous remarks, $\pi _k \widetilde {Y}_{1}^{(k)}f\neq 0$ unless f is not divisible by $x_1$ . Furthermore, the only monomial from $T_1^{-1}f$ that survives is $s_1 f$ . Applying this repeatedly, we obtain

$$ \begin{align*} \pi_k \widetilde{Y}_{1}^{(k)}f=\textbf{t}^k \pi_k\omega^{-1}_{k}\operatorname{\mathrm{pr}}_kT_{k-1}^{-1}\operatorname{\mathrm{pr}}_k (s_{k-2}\dots s_1 f)=W_{1}^{(k)}f. \end{align*} $$

Therefore, $\pi _k\widetilde {Z}_{i}^{(k)} f=0=\widetilde {Z}_{i}^{(k-1)} \pi _k f$ , as expected.

As a consequence, we obtain the following.

Proposition 6.15. For any $i\geq 1$ , the sequence of operators $(\widetilde {Z}_i^{(k)})_{k\geq 2}$ induces a map

$$ \begin{align*} \widetilde{Z}_i^{(\infty)}: \mathscr{P}_{\infty}^+\to \mathscr{P}_{\infty}^{+} \end{align*} $$

such that $\Pi _k \widetilde {Z}_i^{(\infty )}=\widetilde {Z}_i^{(k)}$ for all $k\geq i$ , $k\geq 2$ .

Example 6.16. In continuation of Example 6.12, we illustrate the actions of the operators $W_1^{(k)}$ and $\widetilde {Z}_1^{(k)}$ on the constant sequence $x_2^2$ . We have

$$ \begin{align*} W_1^{(k)}\cdot x_2^2=\textbf{q}^2 \textbf{t}(\textbf{t}-1)x_1^2 \quad \text{ and }\quad \widetilde{Z}_1^{(k)}\cdot x_2^2=\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 (x_1x_3+\cdots +x_1x_k),~ k\geq 3. \end{align*} $$

Clearly, both sequences have inverse limit, in particular,

$$ \begin{align*} \widetilde{Z}_1^{(\infty)}\cdot x_2^2=\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 x_1 e_1[\mathbf{X}_2]. \end{align*} $$

Example 6.17. In continuation of Example 6.13, we illustrate the actions of the operators $W_1^{(k)}$ and $Z_1^{(k)}$ on the sequence $F_k=x^2_2+\cdots +x^2_k$ . We have

$$ \begin{align*} W_1^{(2)}\cdot F_{2}&=W_1^{(2)}x_2^2=\textbf{q}^2(\textbf{t}^2-\textbf{t})x_1^2,\\ W_1^{(3)}\cdot F_{3}&=W_1^{(3)}(x_2^2+x_3^2)=\textbf{q}^2(\textbf{t}^2-\textbf{t})x_1^2+\textbf{q}^2(\textbf{t}^3-\textbf{t}^2)x_1^2= \textbf{q}^2(\textbf{t}^3-\textbf{t})x_1^2, \end{align*} $$

and, more generally, for $k\geq 2$ ,

$$ \begin{align*} W_1^{(k)}\cdot F_{k}=\textbf{q}^2(\textbf{t}^k-\textbf{t})x_1^2. \end{align*} $$

Furthermore,

$$ \begin{align*} \widetilde{Z}_1^{(k)} \cdot F_{k}=\widetilde{Y}_1^{(k)}\cdot F_{k}-W_1^{(k)}\cdot F_{k}=\textbf{q}\textbf{t}(\textbf{t}-1)(x_1 x_2+x_1 x_3+ \ldots + x_1x_k), \end{align*} $$

which can be seen to have inverse limit. Therefore,

$$ \begin{align*} \widetilde{Z}_1^{(\infty)}\cdot F[\mathbf{X}]=\textbf{q}\textbf{t}(\textbf{t}-1)x_1 e_1[\mathbf{X}_1]. \end{align*} $$

6.9

Proposition 6.15 motivates the concept of limit we define as follows; We emphasize that this concept of limit depends intrinsically on the structure of the subspace $\mathscr {P}_{\mathrm {as}}^+\subset \mathscr {P}^+_{\infty }$ .

Let $R(\textbf {t},\textbf {q})=A(\textbf {t},\textbf {q})/B(\textbf {t},\textbf {q})\in \mathbb {Q}(\textbf {t},\textbf {q})$ , with $A(\textbf {t},\textbf {q}),~B(\textbf {t},\textbf {q})\in \mathbb {Q}[\textbf {t},\textbf {q}]$ . The order of vanishing at $\textbf {t}=0$ for $R(\textbf {t},\textbf {q})$ , denoted by

$$ \begin{align*} \operatorname{\mathrm{ord}} R(\textbf{t},\textbf{q}), \end{align*} $$

is the difference between the order of vanishing at $\textbf {t}=0$ for $A(\textbf {t},\textbf {q})$ and $B(\textbf {t},\textbf {q})$ .

We say that the sequence $(a_n)_{n\geq 1}\subset \mathbb {Q}(\textbf {t},\textbf {q})$ converges to $0$ if the sequence $(\operatorname {\mathrm {ord}} a_n)_{n\geq 1}\subset \mathbb Z$ converges to $+\infty $ . We say that the sequence $(a_n)_{n\geq 1}\subset \mathbb {Q}(\textbf {t},\textbf {q})$ converges to a if $(a_n-a)_{n\geq 1}$ converges to $0$ . We write,

$$ \begin{align*} \lim_{n\to \infty} a_n=a. \end{align*} $$

Definition 6.18. Let $(f_k)_{k\geq 1}$ be a sequence with $f_k\in \mathscr {P}^+_k$ . We say that the sequence is convergent if there exists $N\geq 1$ and sequences $(h_k)_{k\geq 1}$ , $(g_{i,k})_{k\geq 1}$ , $i\leq N$ , $h_k, ~g_{i, k}\in \mathscr {P}^+_k$ and $(a_{i, k})_{k\leq 1}$ , $i\leq N$ , $a_{i, k}\in \mathbb {Q}(\textbf {t},\textbf {q})$ such that

1. a) For any $k\geq 1$ , we have $f_k=h_k+\sum _{i=1}^N a_{i, k} g_{i, k}$ ;

2. b) For any $i\leq N$ , $k\geq 2$ , $\pi _k(g_{i, k})=g_{i, k-1}$ and $\pi _k(h_{k})=h_{k-1}$ . We denote by

   $$ \begin{align*} \displaystyle g_i=\lim_{k\to \infty} g_{i,k}\quad \text{and}\quad \displaystyle h=\lim_{k\to \infty} h_{k} \end{align*} $$
   the sequence $(g_{i,k})_{k\geq 1}$ and, respectively, $(h_{k})_{k\geq 1}$ as elements of $\mathscr {P}_{\infty }^+$ . We require that $g_i\in \mathscr {P}_{\mathrm {as}}^+$ .

3. c) For any $i\leq N$ , the sequence $(a_{i, k})_{k\geq 1}$ is convergent. We denote $\displaystyle a_i=\lim _{k\to \infty }(a_{i, k})$ .

If the sequence $(f_k)_{k\geq 1}$ is convergent, we define its limit as

$$ \begin{align*} \lim_k(f_k):=h+ \sum_{i=1}^N a_i g_i\in\mathscr{P}_{\infty}^+. \end{align*} $$

Example 6.19. The sequence

$$ \begin{align*} f_k=(1+\textbf{t}+\ldots +\textbf{t}^k)e_i[\overline{\mathbf{X}}_{k}], \end{align*} $$

has the limit

$$ \begin{align*} \lim_k f_k=\frac{1}{1-\textbf{t}}e_i[\mathbf{X}]. \end{align*} $$

The sequence

$$ \begin{align*} g_k=\textbf{t}^k e_i[\overline{\mathbf{X}}_{k}] \end{align*} $$

has limit $0$ .

We show that the limit of a sequence does not depend on the choice of the auxiliary sequences in Definition 6.18.

Proof. It suffices to show that the limit of the constant sequence $0$ is zero, regardless of the auxiliary sequences in Definition 6.18. Consider sequences $(c_{i,k})_{k\geq 1}$ and $(q_{i,k})_{k\geq 1}$ such that

$$ \begin{align*} 0=\sum_{i=1}^{N}c_{i,k}q_{i,k}\in \mathscr{P}_{k}^{+}, \end{align*} $$

and

$$ \begin{align*} \lim_{k\rightarrow\infty}c_{i,k}=c_i\in \mathbb{Q}(\textbf{t},\textbf{q}),\ \lim_{k\rightarrow\infty}q_{i,k}=q_i\in\mathscr{P}_{\mathrm{as}}^{+}. \end{align*} $$

We need to show that $\sum _{i=1}^N c_iq_i=0$ .

Without loss of generality, we assume that $q_1,\ldots ,q_N$ are $\mathbb {Q}$ -linearly independent. Indeed, any linear relation between $q_1,\ldots ,q_N$ must also hold for $q_{1,k},\ldots ,q_{N,k}$ for all k. We can therefore substitute one of them, say $q_{1,k}$ , with the same $\mathbb {Q}$ -linear combination of $q_{2,k},\ldots ,q_{N,k}$ for all k. It is clear that the conclusion does not change after such a substitution.

Recall that each $q_i\in \mathscr {P}_{\mathrm {as}}^{+}$ . We can find $n\geq 1$ such that $q_i\in \mathscr {P}(n)$ for all $1\leq i\leq N$ . For the same reason as before, without loss of generality, we can assume that

$$ \begin{align*} q_i= f_i(x_1,\ldots ,x_n)e_{\alpha_i}[\mathbf{X}_{n}], \end{align*} $$

where

$$ \begin{align*} \alpha_i=(\alpha_{i,1}\geq\dots \geq\alpha_{i,s_i}), \quad 1\leq i\leq N \end{align*} $$

are distinct partitions. Let

$$ \begin{align*} M=\max_{1\leq i\leq N}\alpha_{i,1}. \end{align*} $$

For any $k>M+n$ , we claim that $q_{1,k},\ldots ,q_{N,k}$ are also $\mathbb {Q}$ -linear independent. Indeed, if we have a linear relation

$$ \begin{align*} \sum_{i=1}^{N}a_i f_i(x_1,\ldots ,x_n)e_{\alpha_i}[x_{n+1},\ldots ,x_{k}]=0 \end{align*} $$

for some $a_i\in \mathbb {Q}$ , then, for any evaluation at $x_1=b_1,\ldots ,x_n=b_n,$ we have

$$ \begin{align*} \sum_{i=1}^{N}a_i f_i(b_1,\ldots ,b_n)e_{\alpha_i}[x_{n+1},\ldots ,x_{k}]=0. \end{align*} $$

Note that, because $k-n>M$ and the partitions $\alpha _i$ are distinct, the symmetric functions $e_{\alpha _i}[\mathbf {X}_{n}]$ are linearly independent. Therefore, for all $1\leq i\leq N$ ,

$$ \begin{align*} a_i f_i(b_1,\ldots ,b_n)=0. \end{align*} $$

Since $f_i$ are polynomials in finitely many variables, we obtain, for all $1\leq i\leq N$ ,

$$ \begin{align*} a_i f_i(x_1,\ldots ,x_n)=0, \end{align*} $$

which is a contradiction. Therefore, for k large enough, we have that $q_{1,k},\ldots ,q_{N,k}$ are $\mathbb {Q}$ -linear independent.

We can now prove that for k large enough, all $c_{i,k}$ will necessarily be $0$ . Indeed, from the hypothesis we have

$$ \begin{align*} 0=\sum_{i=1}^{N}c_{i,k}q_{i,k}. \end{align*} $$

By multiplying both sides the common denominator of $c_{i,k}$ we may assume all $c_{i,k}$ are polynomials in two variables $\textbf {t},\textbf {q}$ . Again, for any evaluation $\textbf {q}=a,\textbf {t}=b$ , we have

$$ \begin{align*} 0=\sum_{i=1}^{N}c_{i,k}(a,b)q_{i,k}. \end{align*} $$

But we already now $q_{1,k},\ldots ,q_{N,k}$ are $\mathbb {Q}$ -linear independent for k large enough. Hence, it forces $c_{i,k}(a,b)=0$ for all i. Again, this implies that $c_{i,k}(\textbf {t},\textbf {q})=0$ , $1\leq i\leq N$ .

6.10

For later use, we record the following result. To set the notation, assume that $A_k:\mathscr {P}_k^+\to \mathscr {P}_k^+$ , $k\geq 1$ , is a sequence of operators with the following property: For any $f\in \mathscr {P}_{\mathrm {as}}^+$ , the sequence $(A_k \Pi _k f)_{k\geq 1}$ converges to an element of $\mathscr {P}_{\mathrm {as}}^+$ . Let ${A}$ be the operator

$$ \begin{align*} A: \mathscr{P}_{\mathrm{as}}^+\to \mathscr{P}_{\mathrm{as}}^+,\quad f\mapsto \lim_k {A}_k \Pi_k f. \end{align*} $$

We refer to A as the limit operator of the sequence $(A_k)_{k\geq 1}$ .

Proposition 6.21. Let $(f_k)_{k\geq 1}$ , $f_k\in \mathscr {P}_k^+$ be a convergent sequence such that $f=\lim _{k} f_k\in \mathscr {P}_{\mathrm {as}}^+$ . Then, with the notation above, we have

$$ \begin{align*} A f=\lim_{k}A_{k} f_k. \end{align*} $$

Proof. By replacing $f_k$ with $f_k-\Pi _k$ , it is enough to prove the statement for the case $f=0$ . Assume that $f=0$ . In this case, there exist sequences $(c_{i,k})_{k\geq 1}$ and $(q_{i,k})_{k\geq 1}$ as in Definition 6.18 such that

$$ \begin{align*} f_k=\sum_{i=1}^{N}c_{i,k}q_{i,k}\in \mathscr{P}_{k}^{+}, \end{align*} $$

with

$$ \begin{align*} \lim_{k\rightarrow\infty}c_{i,k}=c_i\in \mathbb{Q}(\textbf{t},\textbf{q}),\ \lim_{k\rightarrow\infty}q_{i,k}=q_i\in\mathscr{P}_{\mathrm{as}}^{+}. \end{align*} $$

For each i such that $c_i\neq 0$ , we may replace $c_{i,k}$ with $c_{i,k}/c_i$ and $q_{i,k}$ with $c_i q_{i,k}$ . This allows us to assume that $c_i\in \{0,1\}$ . Without loss of generality, we assume that $q_1,\ldots ,q_N$ are $\mathbb {Q}$ -linearly independent. Indeed, any linear relation between $q_1,\ldots ,q_N$ must also hold for $q_{1,k},\ldots ,q_{N,k}$ , for all k. We can therefore substitute one of them, say $q_{1,k}$ , with the same $\mathbb {Q}$ -linear combination of $q_{2,k},\ldots ,q_{N,k}$ for all k. It is clear that the conclusion does not change after such a substitution. It is now clear that for all i we have $c_i=0$ .

For any $k\geq 1$ ,

$$ \begin{align*} A_k f_k=\sum_{i=1}^N c_{i,k} A_k q_{i,k}=\sum_{i=1}^N c_{i,k} A_k \Pi_k q_i. \end{align*} $$

Since $\displaystyle \lim _{k\rightarrow \infty }A_k \Pi _k q_i=A q_i\in \mathscr {P}_{\mathrm {as}}^{+}$ , we have, by Definition 6.18,

$$ \begin{align*} \lim_k A_k f_k=\sum_{i=1}^N c_{i,k} A q_i=0. \end{align*} $$

This is precisely our claim.

This result can be interpreted as a property of continuity for the operator A. Let $(B_k)_{k\geq 1}$ be another sequence of operators with the same property as $(A_k)_{k\geq 1}$ and denote by $B: \mathscr {P}_{\mathrm {as}}^+\to \mathscr {P}_{\mathrm {as}}^+$ the corresponding limit operator.

Proof. Let $f\in \mathscr {P}_{\mathrm {as}}^+$ . Since $\displaystyle \lim _k B_k\Pi _kf=B f\in \mathscr {P}_{\mathrm {as}}^+$ , we can apply Proposition 6.21 to obtain

$$ \begin{align*} \lim_k A_k B_k \Pi_k f=A B f \in \mathscr{P}_{\mathrm{as}}^+, \end{align*} $$

which proves our claim.

6.11

Let us examine the following situation.

Lemma 6.23. Let $f\in \mathscr {P}_{\mathrm {as}}^+$ and $i\geq 1$ . The sequence $W_i^{(k)} \Pi _k f$ converges. We define

$$ \begin{align*} W^{(\infty)}_i: \mathscr{P}_{\mathrm{as}}^+\to \mathscr{P}_{\infty}^+,\quad W^{(\infty)}_i f=\lim_{k} W_i^{(k)}\Pi_k f\in \mathscr{P}_{\infty}^{+}. \end{align*} $$

Proof. It is enough to prove our claim for $W_1$ and $f=g(x_1,\dots ,x_n)m_{\alpha }[\mathbf {X}_n]$ for some $n\geq 1$ , where g is a monomial and $m_{\alpha }$ is the monomial symmetric function in the indicated alphabet. If $x_1$ divides g, then $W_1f=0$ . We assume that $x_1$ does not divide g.

We denote by $\ell (\alpha )$ the length of the partition $\alpha $ . Of course, for any $k\geq n+\ell (\alpha )$ , we have

$$ \begin{align*} \Pi_k m_{\alpha}[\mathbf{X}_n]=m_{\alpha}[x_{n+1},\dots ,x_k]. \end{align*} $$

If $k\geq n+\ell (\alpha )$ , denote by $a_1,\dots , a_s$ the distinct parts of $\alpha $ . Let $\beta _i$ , $1\leq i\leq s$ the partition obtained by eliminating one part of size $a_i$ from $\alpha $ . Then,

$$ \begin{align*} m_{\alpha}[x_{n+1},\dots,x_k]=\sum_{i=1}^s \sum_{j=n+\ell(\alpha)}^k m_{\beta_i}[x_{n+1},\dots, x_{j-1}] x_j^{a_i}. \end{align*} $$

Therefore, we have

$$ \begin{align*} W_{1}^{(k)}\Pi_k f &=g(x_1,\dots,x_n) \sum_{i=1}^s \sum_{j=n+\ell(\alpha)}^k W_1^{(k)} m_{\beta_i}[x_{n+1},\dots, x_{j-1}] x_j^{a_i}\\ &=g(x_1,\dots,x_n) \sum_{i=1}^s \sum_{j=n+\ell(\alpha)}^k (\textbf{t}^j-\textbf{t}^{j-1})\textbf{q}^{a_i} x_1^{a_i} m_{\beta_i}[x_{n+1},\dots, x_{j-1}]. \end{align*} $$

For a monomial $m=x_{i_1}^{\eta _1}\cdots x_{i_M}^{\eta _M}$ , $n<i_1<\dots <i_M$ , $\eta _1,\dots ,\eta _M\geq 1$ , we denote $\ell (m)=i_M$ and we write $m\in [\beta ]$ if $(\eta _1,\dots , \eta _M)$ is a permutation of $\beta $ and we write $m\in [\beta ]_k$ if $m\in [\beta ]$ and $\ell (m)\leq k$ . With this notation, we have

$$ \begin{align*} W_{1}^{(k)}\Pi_k f &=g(x_1,\dots,x_n) \sum_{i=1}^s \sum_{m\in [\beta_i]_k} (\textbf{t}^k-\textbf{t}^{\ell(m)})\textbf{q}^{a_i} x_1^{a_i} m\\ &= g(x_1,\dots,x_n)\left(\sum_{i=1}^s \textbf{t}^k \textbf{q}^{a_i} x_1^{a_i} m_{\beta_i}[x_{n+1},\dots,x_k] -\sum_{i=1}^s \textbf{q}^{a_i} x_1^{a_i} \sum_{m\in [\beta_i]_k} \textbf{t}^{\ell(m)} m \right) \end{align*} $$

According to Definition 6.18,

$$ \begin{align*} \lim_k W_{1}^{(k)}\Pi_k f = - \sum_{i=1}^s \textbf{q}^{a_i} x_1^{a_i} g(x_1,\dots,x_n) \sum_{m\in [\beta_i]} \textbf{t}^{\ell(m)} m \in \mathscr{P}_{\infty}^+, \end{align*} $$

which proves our claim.

6.12

We can now prove the following.

Proposition 6.25. Let $f\in \mathscr {P}_{\mathrm {as}}^+$ and $i\geq 1$ . The sequence $\widetilde {Y}_i^{(k)} \Pi _k f$ converges. We define

$$ \begin{align*} \mathscr{Y}_i: \mathscr{P}_{\mathrm{as}}^+\to \mathscr{P}_{\infty}^+,\quad \mathscr{Y}_i f=\lim_{k} \widetilde{Y}_i^{(k)}\Pi_k f\in \mathscr{P}_{\infty}^{+}. \end{align*} $$

Example 6.26. In continuation of Examples 6.13 and 6.17, we have

$$ \begin{align*} \mathscr{Y}_1 \cdot p_2[\mathbf{X}_1]=-\textbf{q}^2 \textbf{t} x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)x_1 e_1[\mathbf{X}_1]. \end{align*} $$

As it can be seen, $p_2[\mathbf {X}_1]$ and $\mathscr {Y}_1 \cdot p_2[\mathbf {X}_1]$ are elements of $\mathscr {P}(1)^+$ .

Also, from Examples 6.12 and 6.16 we have

$$ \begin{align*} \mathscr{Y}_1 \cdot x_2^2=\textbf{q}^2 \textbf{t}(\textbf{t}-1)x_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)x_1 x_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 x_1 e_1[\mathbf{X}_2]. \end{align*} $$

In this case, $x_2^2$ and $\mathscr {Y}_1 x_2^2$ are elements of $\mathscr {P}(2)^+$ .

It turns out that the image of $\mathscr {Y}_i$ is contained in $\mathscr {P}_{\mathrm {as}}^+$ . We will use the following result.

Lemma 6.27. Let $f\in \mathbb {Q}(\textbf {t},\textbf {q})[x_{k+1},x_{k+2},\dots ]$ be a polynomial satisfying

$$ \begin{align*} T_i f=f,\quad \textrm{ for }k+1\leq i\leq k+m. \end{align*} $$

Let $g=T_{k+m}^{-1}\dots T_{k+1}^{-1}T_k^{-1}(x_{k}^{s}f)$ for some $s\geq 0$ . Then,

$$ \begin{align*} T_i g=g \quad \textrm{ for }k\leq i\leq k+m-1. \end{align*} $$

Lemma 6.28. $\mathscr {P}(k)^{+}$ is stable under the action of $\mathscr {Y}_1$ . Therefore, we have

$$ \begin{align*} \mathscr{Y}_i:\mathscr{P}_{\mathrm{as}}^{+}\rightarrow \mathscr{P}_{\mathrm{as}}^{+}. \end{align*} $$

Proof. Let $f\in \mathscr {P}(k)^{+}$ (recall the definition of $\mathscr {P}(k)^{+}$ in §2.6). For any $m>k+1$ , we have

$$ \begin{align*} \widetilde{Y}_1^{(m)} \Pi_m f=\textbf{t}^{m} \varpi_m T_{m-1}^{-1}\dots T_{1}^{-1}\Pi_m f. \end{align*} $$

Since $f\in \mathscr {P}(k)^{+}$ , $\Pi _m f$ is fixed under the action of $T_{k+1},\dots ,T_{m-1}$ . Now, we have

$$ \begin{align*} \varpi_m T_{m-1}^{-1}\dots T_{1}^{-1}\Pi_m f=\varpi_m T_{m-1}^{-1}\dots T_{k+1}^{-1}T_{k}^{-1}(T_{k-1}^{-1}\dots T_{1}^{-1}\Pi_m f),\end{align*} $$

and we may write

$$ \begin{align*} T_{k-1}^{-1}\dots T_{1}^{-1}\Pi_m f=\sum_{j} c_j[x_1,\dots ,x_{k-1}]x_{k}^{s_j}f_j \end{align*} $$

as a finite sum, where each $f_j$ is a polynomial in $\mathbb {Q}(\textbf {t},\textbf {q})[x_{k+1},x_{k+2},\dots ]$ satisfying

$$ \begin{align*} T_i f_j=f_j,\quad \textrm{ for }i=k+1,k+2,\dots ,m-1. \end{align*} $$

Applying Lemma 6.27 for $T_{k-1}^{-1}\dots T_{1}^{-1}\Pi _m f$ , we obtain that $ T_{m-1}^{-1}\dots T_{1}^{-1}\Pi _m f$ is fixed under the action of the elements $T_{k},\dots ,T_{m-2}$ . The relation (6.3) implies that $\varpi _m T_{m-1}^{-1}\dots T_{1}^{-1}\Pi _m f$ is fixed under the action of $T_{k+1},\dots ,T_{m-1}$ . Therefore, $\widetilde {Y}_1^{(m)} \Pi _m f$ is symmetric in $x_{k+1},\dots , x_m$ for all $m>k+1$ . In conclusion, the limit

$$ \begin{align*} \displaystyle \lim_{m} \widetilde{Y}_i^{(m)}\Pi_m f\in \mathscr{P}(k)^+, \end{align*} $$

proving our claim.

We can now state the following.

As we will show in Theorem 6.34, these operators define an action of $\mathscr {H}^+$ on $\mathscr {P}_{\mathrm {as}}^+$ .

6.13

In fact, we can obtain a more precise description of the action of $\mathscr {Y}_1$ on $\mathscr {P}(k)^+$ . First, let us record the following technical result. We use the notation in §2.3.

Lemma 6.31. Let $x_1^{n}\in \mathscr {P}^+_{m+1}$ . Then,

$$ \begin{align*} \textbf{t}^{m}T_{m}^{-1}\ldots T_{1}^{-1}x_1^{n}=\sum_{i=0}^{n-1}x_{m+1}^{n-i}h_i[(1-\textbf{t})\overline{\mathbf{X}}_{m}]. \end{align*} $$

Proof. We will prove the result by induction on m. By direct computation, we obtain

$$ \begin{align*} T_{1}^{-1}x_1^{n}&=\textbf{t}^{-1}x_2^{n}+(\textbf{t}^{-1}-1)(x_2^{n-1}x_1+x_2^{n-2}x_1^2+\ldots +x_2x_1^{n-1})\\ &=\textbf{t}^{-1}\sum_{i=0}^{n-1}x_{2}^{n-i}h_i[(1-\textbf{t})x_1]. \end{align*} $$

Assume that our claim holds for $m-1$ . Then, we have

$$ \begin{align*} \textbf{t}^{m}T_{m}^{-1}\ldots T_{1}^{-1}x_1^{n}&=\textbf{t} T_{m}^{-1}\sum_{i=0}^{n-1}x_{m}^{n-i}h_i[(1-\textbf{t})\overline{\mathbf{X}}_{m-1}]\\ &=\sum_{i=0}^{n-1}\sum_{j=0}^{n-i-1}x_{m+1}^{n-i-j}h_j[(1-\textbf{t})x_m] h_i[(1-\textbf{t})\overline{\mathbf{X}}_{m-1}]\\ &=\sum_{l=0}^{n-1} x_{m+1}^{n-l}\sum_{\substack{i+j=l\\i, j\geq 0}}h_j[(1-\textbf{t})x_m] h_i[(1-\textbf{t})\overline{\mathbf{X}}_{m-1}]\\ &=\sum_{l=0}^{n-1} x_{m+1}^{n-l}h_l[(1-\textbf{t})\overline{\mathbf{X}}_{m}], \end{align*} $$

as expected.

Recall that for all k, the multiplication map $ \mathscr {P}_k^{+}\otimes \operatorname {\mathrm {Sym}}[\mathbf {X}_k] \cong \mathscr {P}(k)^+$ is an algebra isomorphism.

Proposition 6.32. Let $n\geq 0$ , $f(x_1,\dots ,x_{k-1})\in \mathscr {P}_{k-1}^{+}$ , and $G[\mathbf {X}_{k-1}]\in \operatorname {\mathrm {Sym}}[\mathbf {X}_{k-1}]$ . We regard

$$ \begin{align*} F=f(x_1,\dots,x_{k-1})x_k^n G[\mathbf{X}_{k-1}] \end{align*} $$

as an element of $\mathscr {P}(k)^+$ . Then,

$$ \begin{align*} \mathscr{Y}_1\mathscr{T}_1\cdots \mathscr{T}_{k-1} F = \frac{\textbf{t}^k}{1-\textbf{t}} f(x_2,\dots,x_{k})G[\mathbf{X}_{k}+\textbf{q} x_1](h_n[(1-\textbf{t} )(\mathbf{X}_k+\textbf{q} x_1)]-h_n[(1-\textbf{t} )\mathbf{X}_k]). \end{align*} $$

Proof. For any $m>k+1$ , we have

$$ \begin{align*} \widetilde{Y}_1^{(m)} T_1\cdots T_{k-1} \Pi_m F&=\textbf{t}^{m} \varpi_m T_{m-1}^{-1}\dots T_{k}^{-1} f(x_1,\dots,x_{k-1})x_k^n G[\overline{\mathbf{X}}_{[k,m]}]\\ &=\textbf{t}^{k} \varpi_m f(x_1,\dots,x_{k-1})G[\overline{\mathbf{X}}_{[k,m]}] \textbf{t}^{m-k} T_{m-1}^{-1}\dots T_{k}^{-1} x_k^n\\ &=\textbf{t}^{k} \varpi_m f(x_1,\dots,x_{k-1})G[\overline{\mathbf{X}}_{[k,m]}] \sum_{i=0}^{n-1}x_{m}^{n-i}h_i[(1-\textbf{t})\overline{\mathbf{X}}_{[k,m-1]}]\\ &=\textbf{t}^{k} f(x_2,\dots,x_{k})G[\overline{\mathbf{X}}_{[k+1,m]}+\textbf{q} x_1] \sum_{i=0}^{n-1}(\textbf{q} x_{1})^{n-i}h_i[(1-\textbf{t})\overline{\mathbf{X}}_{[k+1,m]}]. \end{align*} $$

Therefore, $\displaystyle \lim _{m}\widetilde {Y}_i^{(m)}T_1\cdots T_{k-1} \Pi _m F$ , equals

$$ \begin{align*} \textbf{t}^{k} f(x_2,\dots,x_{k})& G[\mathbf{X}_{k}+\textbf{q} x_1] \sum_{i=0}^{n-1}(\textbf{q} x_{1})^{n-i}h_i[(1-\textbf{t}){\mathbf{X}}_{k}]\\ &=\frac{\textbf{t}^{k}}{1-\textbf{t}} f(x_2,\dots,x_{k})G[{\mathbf{X}}_{k}+\textbf{q} x_1](h_n[(1-\textbf{t} )(\mathbf{X}_k+\textbf{q} x_1)]-h_n[(1-\textbf{t} )\mathbf{X}_k]), \end{align*} $$

proving our claim.

6.14

We establish the following result in preparation for the proof of Theorem 6.34.

Lemma 6.33. Let $f\in \mathscr {P}_{\mathrm {as}}^+$ . Then,

$$ \begin{align*} \lim_{k} [\widetilde{Y}_{i}^{(k)},\widetilde{Y}_{j}^{(k)}]\Pi_k f=0. \end{align*} $$

Proof. First note that we can apply the following two relations recursively:

$$ \begin{align*} [\widetilde{Y}_{i}^{(k)},\widetilde{Y}_{j}^{(k)}]=\textbf{t}^{-1}T_{j-1}[\widetilde{Y}_{i}^{(k)},\widetilde{Y}_{j-1}^{(k)}]T_{j-1},\quad \textrm{for } i>j, \end{align*} $$

$$ \begin{align*} [\widetilde{Y}_{1}^{(k)},\widetilde{Y}_{i}^{(k)}]=\textbf{t}^{-1}T_{i-1}[\widetilde{Y}_{1}^{(k)},\widetilde{Y}_{i-1}^{(k)}]T_{i-1},\quad \textrm{for } i>2. \end{align*} $$

Hence, it suffices to prove the result for $[\widetilde {Y}_{1}^{(k)},\widetilde {Y}_{2}^{(k)}]$ .

By Remark 6.7, we have

$$ \begin{align*} [\widetilde{Y}_{1}^{(k)},\widetilde{Y}_{2}^{(k)}]=\textbf{t}^{2k-1}\gamma_k T_{k-1}^{-1}\ldots T_1^{-1}T_{k-1}^{-1}\ldots T_2^{-1}=\textbf{t}^{2k-1}\gamma_k T_{k-2}^{-1}\ldots T_1^{-1}T_{k-1}^{-1}\ldots T_1^{-1}. \end{align*} $$

Let $f(x_1,\ldots ,x_m)F[\mathbf {X}]\in \mathscr {P}(k)^{+}$ , where $F[\mathbf {X}]$ is fully symmetric and nonzero. Then for $k>m$ , we have

$$ \begin{align*} \gamma_k T_{k-1}^{-1}\ldots T_1^{-1}T_{k-1}^{-1}\ldots T_2^{-1}\Pi_k f(x_1,\ldots ,x_m)F[\mathbf{X}] &= \gamma_k T_{k-1}^{-1}\ldots T_1^{-1}T_{k-1}^{-1}\ldots T_2^{-1} f(x_1,\ldots ,x_m)F[\overline{\mathbf{X}}_{k}]\\ &=\gamma_k F[\overline{\mathbf{X}}_{k}](T_{k-1}^{-1}\ldots T_1^{-1}T_{k-1}^{-1}\ldots T_2^{-1} f(x_1,\ldots ,x_m))\\ &=0. \end{align*} $$

For the last equality we used equation (6.6). Therefore, in this case,

$$ \begin{align*} \lim_{k\rightarrow\infty} [\widetilde{Y}_{i}^{(k)},\widetilde{Y}_{j}^{(k)}]\Pi_k f(x_1,\ldots ,x_m)F[\mathbf{X}]=0. \end{align*} $$

It remains to compute the limit for $f(x_1,\ldots ,x_m)\in \mathscr {P}_m^+$ . Let $k>m+1$ . Without loss of generality, we may assume that

$$ \begin{align*} T_{m-2}^{-1}\ldots T_1^{-1}T_{m-1}^{-1}\ldots T_1^{-1} f(x_1,\ldots ,x_m) \end{align*} $$

is a monomial of the form $x_m^{n}x_{m-1}^{n^{\prime }}g(x_1,\ldots ,x_{m-2})$ . If $n>0$ , we have

$$ \begin{align*} [\widetilde{Y}_{1}^{(k)},\widetilde{Y}_{2}^{(k)}] f(x_1,\ldots ,x_m) &= \gamma_k T_{k-2}^{-1}\ldots T_1^{-1}T_{k-1}^{-1}\ldots T_1^{-1} f(x_1,\ldots ,x_m)\\ &=\gamma_k T_{k-2}^{-1}\ldots T_{m-1}^{-1}T_{k-1}^{-1}\ldots T_{m}^{-1}(T_{m-2}^{-1}\ldots T_1^{-1}T_{m-1}^{-1}\ldots T_1^{-1} f(x_1,\ldots ,x_m))\\ &=\gamma_k T_{k-2}^{-1}\ldots T_{m-1}^{-1}T_{k-1}^{-1}\ldots T_{m}^{-1}x_m^{n}x_{m-1}^{n^{\prime}}g(x_1,\ldots ,x_{m-2})\\ &= g(x_1,\ldots ,x_{m-2}) \gamma_k T_{k-2}^{-1}\ldots T_{m-1}^{-1} x_{m-1}^{n^{\prime}} T_{k-1}^{-1}\ldots T_{m}^{-1}x_m^{n}. \end{align*} $$

If $n=n^{\prime }=0$ , then $[\widetilde {Y}_{1}^{(k)},\widetilde {Y}_{2}^{(k)}]f(x_1,\ldots ,x_m)=0$ by equation (6.6). If $n=0$ and $n^{\prime }>0$ , then by Lemma 6.31

$$ \begin{align*} \gamma_k T_{k-2}^{-1}\ldots T_{m-1}^{-1}x_{m-1}^{n^{\prime}} &= \textbf{t}^{m-k}\gamma_k \sum_{i=0}^{n^{\prime} -1} x_{k-1}^{n^{\prime} -i} h_i[(1-\textbf{t})\overline{\mathbf{X}}_{[m-1,k-2]}]\\ &= \textbf{t}^{m-k} \sum_{i=0}^{n^{\prime} -1} h_i[(1-\textbf{t})\overline{\mathbf{X}}_{{[m+1,k]}}]\gamma_k x_{k-1}^{n^{\prime} -i} \\ &=\textbf{t}^{m-k}(1-\textbf{t}) \sum_{i=0}^{n^{\prime} -1} {\textbf{q}^{n^{\prime}-i}}h_i[(1-\textbf{t})\overline{\mathbf{X}}_{{[m+1,k]}}](x_1^{n^{\prime}-i-1}x_2+\cdots+x_1x_2^{n^{\prime}-i-1}). \end{align*} $$

Therefore, $\displaystyle \lim _{k} [\widetilde {Y}_{1}^{(k)},\widetilde {Y}_{2}^{(k)}]f(x_1,\ldots ,x_m)=0$ .

If $n^{\prime }=0$ and $n>0$ , then by Lemma 6.31

$$ \begin{align*} \gamma_k T_{k-2}^{-1}\ldots T_{m-1}^{-1}T_{k-1}^{-1}\ldots T_{m}^{-1}x_{m}^{n} &= \textbf{t}^{m-k}\gamma_k T_{k-2}^{-1}\ldots T_{m-1}^{-1} \sum_{i=0}^{n -1} x_k^{n -i} h_i[(1-\textbf{t})\overline{\mathbf{X}}_{[m,k-1]}]. \end{align*} $$

By writing

$$ \begin{align*} h_i[(1-\textbf{t})\overline{\mathbf{X}}_{[m,k-1]}]=\sum_{j=0}^i h_{i-j}[(1-\textbf{t})\overline{\mathbf{X}}_{[m-1,k-1]}]h_j[(\textbf{t}-1)x_{m-1}], \end{align*} $$

and using again Lemma 6.31 for $T_{k-2}^{-1}\ldots T_{m-1}^{-1} x_{m-1}^j$ as well as equation (6.6), we write $\gamma _k T_{k-2}^{-1}\ldots T_{m-1}^{-1}T_{k-1}^{-1}\ldots T_{m}^{-1}x_{m}^{n}$ as

$$ \begin{align*} \textbf{t}^{m-k} \sum_{i=0}^{n -1} \gamma_k x_k^{n -i} &h_{i}[(1-\textbf{t})\overline{\mathbf{X}}_{[m-1,k-1]}]\\ &= \textbf{t}^{m-k}(\textbf{t} -1) \sum_{i=0}^{n -1} {\textbf{q}^{n-i}} h_i[(1-\textbf{t})\overline{\mathbf{X}}_{{[m+1,k]}}](x_1^{n-i-1}x_2+\cdots+x_1x_2^{n-i-1}). \end{align*} $$

Therefore, $\displaystyle \lim _{k} [\widetilde {Y}_{1}^{(k)},\widetilde {Y}_{2}^{(k)}]f(x_1,\ldots ,x_m)=0$ .

For the last case, $n,n^{\prime }>0$ , proceeding as in the previous case we obtain that

$$ \begin{align*} \gamma_k T_{k-2}^{-1}\ldots T_{m-1}^{-1} x_{m-1}^{n^{\prime}} T_{k-1}^{-1}\ldots T_{m}^{-1}x_{m}^{n} \end{align*} $$

equals

$$ \begin{align*} \textbf{t}^{m-k}\gamma_k T_{k-2}^{-1}\ldots T_{m-1}^{-1}x_{m-1}^{n^{\prime}} \sum_{i=0}^{n -1} x_k^{n -i} \sum_{j=0}^i h_{i-j}[(1-\textbf{t})\overline{\mathbf{X}}_{[m-1,k-1]}]h_j[(\textbf{t}-1)x_{m-1}]. \end{align*} $$

Lemma 6.31 for $T_{k-2}^{-1}\ldots T_{m-1}^{-1} x_{m-1}^j$ and equation (6.6) imply that $[\widetilde {Y}_{1}^{(k)},\widetilde {Y}_{2}^{(k)}] f(x_1,\ldots ,x_m)=0$ .

6.15

We are now ready to prove our main result.

Proof. The relations that hold in the algebra $\widetilde {\mathscr {H}}_k$ also hold for the corresponding limit operators by the repeated application of Corollary 6.22. Recall that by Remark 6.6 the relation (4.1e) and the first two relations in equation (4.1d) also hold inside the algebras $\widetilde {\mathscr {H}}_k$ . The only relations that do not transfer directly from those in $\widetilde {\mathscr {H}}_k$ are the commutation relations between $\mathscr {Y}_i$ and $\mathscr {Y}_j$ . However, we do have

$$ \begin{align*} [\widetilde{Y}^{(k)}_{i},\widetilde{Y}^{(k)}_{j}]=\textbf{t}^{-1}T_{j-1}[\widetilde{Y}^{(k)}_{i},\widetilde{Y}^{(k)}_{j-1}]T_{j-1}, \quad i>j,\\ [\widetilde{Y}^{(k)}_{1},\widetilde{Y}^{(k)}_{i}]=\textbf{t}^{-1}T_{i-1}[\widetilde{Y}^{(k)}_{1},\widetilde{Y}^{(k)}_{i-1}]T_{i-1}, \quad i>2. \end{align*} $$

These, by application of Corollary 6.22, imply the same relations for the limit operators

$$ \begin{align*} [\mathscr{Y}_{i},\mathscr{Y}_{j}]&=\textbf{t}^{-1}\mathscr{T}_{j-1}[\mathscr{Y}_{i},\mathscr{Y}_{j-1}]\mathscr{T}_{j-1}, \quad i>j,\\ [\mathscr{Y}_{1},\mathscr{Y}_{i}]&=\textbf{t}^{-1}\mathscr{T}_{i-1}[\mathscr{Y}_{1},\mathscr{Y}_{i-1}]\mathscr{T}_{i-1}, \quad i>2. \end{align*} $$

Therefore, for any $i,j$ , the commutativity of $\mathscr {Y}_i$ and $\mathscr {Y}_j$ would follow from the commutativity of $\mathscr {Y}_1$ and $\mathscr {Y}_2$ . Fix $f\in \mathscr {P}_{\mathrm {as}}^+$ . Then, by Proposition 6.21 and Corollary 6.22,

$$ \begin{align*} [\mathscr{Y}_1,\mathscr{Y}_2]f=\lim_k [\widetilde{Y}^{(k)}_{1},\widetilde{Y}^{(k)}_{2}]\Pi_k f, \end{align*} $$

which is $0$ by Lemma 6.33.

We call this representation the standard representation of $\mathscr {H}^+$ . As with the standard representation of $\mathscr {H}^-$ , we expect this representation to be faithful.

7.1

The main result of this section is an explicit connection between the standard representation of $\mathscr {H}^+$ and the double Dyck path algebra and its standard representation. The double Dyck path algebra, denoted in the literature by $\mathbb {A}_{\textbf {q},\textbf {t}}$ is an algebraic structure discovered by Carlsson and Mellit. It plays a critical role in the proof [Reference Carlsson and Mellit6] of the compositional shuffle conjecture [Reference Haglund, Haiman, Loehr, Remmel and Ulyanov14, Reference Haglund, Morse and Zabrocki15] and the proof [Reference Mellit24] of the more general compositional $(km,kn)$ -shuffle conjecture [Reference Gorsky and Neguţ12, Reference Bergeron, Garsia, Sergel Leven and Xin2].

To facilitate the comparison, we will switch the role of the parameters $\textbf {q}$ and $\textbf {t}$ in the original definition of the double Dyck path algebra. Therefore, the definitions below correspond to $\mathbb {A}_{\textbf {t},\textbf {q}}$ . We will first define two quivers $\dot {\textbf {Q}}$ and $\ddot {\textbf {Q}}$ and introduce some conventions.

The quiver $\dot {\textbf {Q}}$ is defined to be the quiver with vertex set $\mathbb {Z}_{\geq 0}$ , and, for all $k\in \mathbb {Z}_{\geq 0}$ , arrows $d_{+}$ from k to $k+1$ , arrows $d_{-}$ from $k+1$ to k and for $k\geq 2$ loops $T_1,\ldots ,T_{k-1}$ from k to k. Note that, to keep the notation as simple as possible, the same label is used to denote many arrows. To eliminate the possible confusion, we adopt the following convention.

Then quiver $\ddot {\textbf {Q}}$ is the quiver with vertex set $\mathbb {Z}_{\geq 0}$ , and, for all $k\in \mathbb {Z}_{\geq 0}$ , arrows $d_{+}$ and $d_{+}^{*}$ from k to $k+1$ , arrows $d_{-}$ from $k+1$ to k and loops $T_1,\ldots ,T_{k-1}$ from k to k. We will adopt the same labelling convention for paths in $\ddot {\textbf {Q}}$ .

Definition 7.2. The Dyck path algebra $\mathbb {A}_{\textbf {t}}$ is defined as the quiver path algebra of $\dot {\textbf {Q}}$ modulo the following relations:

(7.1a) $$ \begin{align} \begin{gathered} T_{i}T_{j}=T_{j}T_{i}, \quad |i-j|>1,\\ T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}, \quad 1\leq i\leq k-2, \end{gathered} \end{align} $$

(7.1b) $$ \begin{align} (T_{i}-1)(T_{i}+\textbf{t})=0, \quad 1\leq i\leq k-1, \end{align} $$

(7.1c) $$ \begin{align} d_-^2T_{k-1}=d_-^2,\quad T_id_-=d_-T_i, \quad 1\leq i\leq k-2. \end{align} $$

(7.1d) $$ \begin{align} T_1 d_{+}^2 = d_{+}^2,\quad d_{+}T_i = T_{i+1}d_{+}, \quad 1\leq i\leq k-1, \end{align} $$

(7.1e) $$ \begin{align} \begin{gathered} d_{-}[d_{+},d_{-}]T_{k-1} = \textbf{t}[d_{+},d_{-}]d_{-}, \quad (k\geq 2)\\ T_{1}[d_{+},d_{-}]d_{+} = \textbf{t} d_{+}[d_{+},d_{-}],\quad (k\geq 1). \end{gathered} \end{align} $$

Definition 7.3. The double Dyck path algebra $\mathbb {A}_{\textbf {t},\textbf {q}}$ is defined as the quiver path algebra of $\ddot {\textbf {Q}}$ modulo the relations (7.1a), (7.1b), (7.1d) and (7.1e) and the following additional relations:

(7.2a) $$ \begin{align} \begin{gathered} T_1 (d_{+}^{*})^2 = (d_{+}^{*})^2,\quad d_{+}^{*}T_i = T_{i+1}d_{+}^{*},\quad 1\leq i\leq k-1, \end{gathered} \end{align} $$

(7.2b) $$ \begin{align} \begin{gathered} \textbf{t} d_{-}[d_{+}^{*},d_{-}] = [d_{+}^{*},d_{-}]d_{-}T_{k-1},\quad (k\geq 2)\\ \textbf{t}[d_{+}^{*},d_{-}]d_{+}^{*} = T_{1}d_{+}^{*}[d_{+}^{*},d_{-}]\quad (k\geq 1), \end{gathered} \end{align} $$

(7.2c) $$ \begin{align} d_{+}z_{i}=z_{i+1}d_{+},\quad d_{+}^{*}y_{i}=y_{i+1}d_{+}^{*},\quad 1\leq i\leq k-1, \end{align} $$

(7.2d) $$ \begin{align} z_1 d_{+}=-\textbf{q}\textbf{t}^{k+1}y_1d_{+}^{*}. \end{align} $$

The notation $y_i$ and $z_i$ refers to the following operators corresponding to loops at k, $1\leq i \leq k$ :

$$ \begin{align*} \begin{gathered} y_1=\frac{1}{\textbf{t}^{k-1}(\textbf{t}-1)}[d_{+},d_{-}]T_{k-1}\cdots T_1,\\[2pt] y_{i+1}=\textbf{t} T_{i}^{-1}y_i T_{i}^{-1},\quad 1\leq i \leq k-1,\\[2pt] z_1=\frac{\textbf{t}^k}{1-\textbf{t}}[d_{+}^{*},d_{-}]T_{k-1}^{-1}\cdots T_1^{-1},\\[2pt] z_{i+1}=\textbf{t}^{-1}T_i z_i T_i,\quad 1\leq i \leq k-1. \end{gathered} \end{align*} $$

Remark 7.4. There exists an $\mathbb Q$ -algebra involution of $\mathbb {A}_{\textbf {t},\textbf {q}}$ defined by

$$ \begin{align*}T_i^{-1}\mapsto T_{i},\ d_{-}\mapsto d_{-},\ d_{+}^{*}\mapsto d_{+},\ \ d_{+}\mapsto d_{+}^{*}\end{align*} $$

and inverts the parameters $\textbf {t}$ and $\textbf {q}$ . We regard $\mathbb {A}_{\textbf {t}}$ as a subalgebra of $\mathbb {A}_{\textbf {t},\textbf {q}}$ and denote its image under the involution by $\mathbb {A}_{\textbf {t}^{-1}}$ .

7.2

For the following result, we refer to [Reference Carlsson and Mellit6, Lemma 5.5] (see also [Reference Mellit24, Proposition 3.1]).

Proposition 7.5. The loops $y_i$ ’s and $z_i$ ’s satisfy the following relations:

(7.3a) $$ \begin{align} \begin{split} y_i T_j &= T_j y_i,\quad i\neq j,j+1 \\ y_{i+1} &=\textbf{t} T_{i}^{-1}y_i T_{i}^{-1},\quad 1\leq i \leq k-1,\\ y_i y_j &= y_j y_i, \quad 1\leq i,j\leq k, \end{split} \end{align} $$

(7.3b) $$ \begin{align} \begin{split} y_i d_{-} &= d_{-} y_i,\quad 1\leq i\leq k-1 \\ d_{+} y_i &= T_1\ldots T_i y_i T_{i}^{-1}\ldots T_1^{-1}d_{+}, \quad 1\leq i\leq k, \end{split} \end{align} $$

(7.3c) $$ \begin{align} \begin{split} z_i T_j &= T_j z_i,\quad i\neq j,j+1 \\ z_{i+1} &= \textbf{t}^{-1}T_i z_i T_i,\quad 1\leq i \leq k-1,\\ z_i z_j &= z_j z_i, \quad 1\leq i,j\leq k, \end{split} \end{align} $$

(7.3d) $$ \begin{align} \begin{split} z_i d_{-} &= d_{-} z_i,\quad 1\leq i\leq k-1 \\ d_{+}^{*} z_i &= T_{1}^{-1}\ldots T_i^{-1} z_i T_i\ldots T_1 d_{+}^{*}, \quad 1\leq i\leq k. \end{split} \end{align} $$

7.3

The algebra $\mathbb {A}_{\textbf {t},\textbf {q}}$ comes with a canonical quiver representation. The representation emerged first from the analysis of certain operators acting on generating functions of Dyck paths. The algebra itself is an abstract formalization of the properties of the relevant operators. We will use the representation defined in [Reference Mellit24, Proposition 3.2, Proposition 3.3]. The relationship between this action and the original action defined in [Reference Carlsson and Mellit6] is explained in [Reference Mellit24, §3.3].

Let $V_k=\mathbb {Q}(\textbf {t},\textbf {q})[y_1,\ldots ,y_k]\otimes \textrm {Sym}[\mathbf {X}]$ , and denote $V_{\bullet }=(V_k)_{k\geq 0}$ . It is important to note that $V_{\bullet }$ is naturally equipped with a $\lambda $ -ring structure. The symmetric group $S_k$ acts on $V_k$ by permuting the variables $y_i$ . We denote by $\zeta _k$ the algebra morphism that acts trivially $\operatorname {\mathrm {Sym}}[\mathbf {X}]$ and acts on $\mathbb {Q}(\textbf {t},\textbf {q})[y_1,\ldots ,y_k]$ as

$$ \begin{align*}\zeta_k f(y_1,\dots,y_{k-1},y_k)=f(y_2,\dots,y_k,\textbf{q} y_1). \end{align*} $$

Finally, we denote by $\mathfrak {c}_{y_i} F$ the constant term of F with respect to $y_i$ .

Theorem 7.7. The following operators define a quiver representation of $\mathbb {A}_{\textbf {t},\textbf {q}}$ on $V_{\bullet }$ :

(7.4) $$ \begin{align} \begin{split} T_i F &= s_i F+(1-\textbf{t})y_i\frac{F-s_i F}{y_i-y_{i+1}}, \\ d_{-}F &= \mathfrak{c}_{y_k}(F[\mathbf{X}-(\textbf{t}-1)y_k]\text{Exp}[-y_k^{-1}\mathbf{X}]), \\ d_{+}F &= -T_1\ldots T_{k}(y_{k+1}F[\mathbf{X}+(\textbf{t}-1)y_{k+1}],\\ d_{+}^{*}F &= \zeta_k F[\mathbf{X}+(\textbf{t}-1)y_{k+1}]. \end{split} \end{align} $$

The formulas above represent the actions of the arrows originating at node k.

We call this representation the standard representation of the double Dyck path algebra $\mathbb {A}_{\textbf {t},\textbf {q}}$ .

An important result is the computation of the action of the loops $y_i$ in the standard representation. We refer to [Reference Carlsson and Mellit6, Lemma 5.4, Lemma 5.5] (see also [Reference Mellit24, Proposition 3.2]).

The action of the $z_i$ operators is more complicated. We record below some examples that show in particular that the action of $z_1$ is not compatible with the canonical inclusion $V_k\subset V_{k+1}$ .

Example 7.9. Let $y_2^2\in V_2$ . We have,

$$ \begin{align*}z_1 \cdot y_2^2 =\textbf{q}^2 \textbf{t}(\textbf{t}-1)y_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)y_1 y_2-\textbf{q}\textbf{t}(\textbf{t}-1)y_1 e_1[\mathbf{X}].\end{align*} $$

Let $y_2^2\in V_3$ . We have,

$$ \begin{align*}z_1 \cdot y_2^2 =\textbf{q}^2 \textbf{t}(\textbf{t}-1)y_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)y_1 y_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 y_1y_3-\textbf{q}\textbf{t}(\textbf{t}-1)y_1 e_1[\mathbf{X}].\end{align*} $$

More generally, for $y_2^2\in V_k$ , $k\geq 3$ , we have,

$$ \begin{align*}z_1 \cdot y_2^2 =\textbf{q}^2 \textbf{t}(\textbf{t}-1)y_1^2+\textbf{q}\textbf{t}(\textbf{t}-1)y_1 y_2-\textbf{q}\textbf{t}(\textbf{t}-1)^2 (y_1y_3+\cdots+y_1y_k)-\textbf{q}\textbf{t}(\textbf{t}-1)y_1 e_1[\mathbf{X}].\end{align*} $$

Example 7.10. Let $p_2[\mathbf {X}]\in V_1$ . We have,

$$ \begin{align*}z_1 \cdot p_2[\mathbf{X}]=\textbf{q}^2 \textbf{t}(1-\textbf{t}^2) y_1^2-\textbf{q}\textbf{t}(1-\textbf{t}^2)y_1 e_1[\mathbf{X}]. \end{align*} $$

Let $p_2[\mathbf {X}]\in V_2$ . We have,

$$ \begin{align*}z_1 \cdot p_2[\mathbf{X}]= \textbf{q}^2 \textbf{t}^2(1-\textbf{t}^2)y_1^2 - \textbf{q}\textbf{t}^2(1-\textbf{t}^2)y_1 e_1[\mathbf{X}]. \end{align*} $$

7.4

We will use the standard representation of $\mathscr {H}^+$ to construct a quiver representation of $\mathbb {A}_{\textbf {t},\textbf {q}}$ . Let

$$ \begin{align*} \mathscr{P}_{\bullet}=(\mathscr{P}(k)^+)_{k\geq 0}. \end{align*} $$

For $k\geq 0$ , recall that we denote by $\mathscr {H}^+(k)$ the subalgebra of $\mathscr {H}^+$ generated by $T_i$ , $1\leq i\leq k-1$ , and $X_i$ , and $Y_i$ , $1\leq i\leq k$ . From Lemma 6.28, we know that each $\mathscr {P}(k)^+$ is stable under the action of $\mathscr {H}^+(k)$ through the standard representation of $\mathscr {H}^+$ . Recall that, for all k, the multiplication map $ \mathscr {P}_k^{+}\otimes \operatorname {\mathrm {Sym}}[\mathbf {X}_k] \cong \mathscr {P}(k)^+$ is an algebra isomorphism.

The elements

$$ \begin{align*} \widetilde{\omega}_k^{-1}, \omega^{-1}_{k}\in \mathscr{H}_k^+ \end{align*} $$

act on $\mathscr {P}_k^+$ via the standard representation of $\mathscr {H}_k^+$ (see Proposition 3.9). We extend their action to $\mathscr {P}(k)^+$ as $\operatorname {\mathrm {Sym}}[\mathbf {X}_k]$ -linear maps. Let

$$ \begin{align*} \iota(k):\mathscr{P}(k)^+\to\mathscr{P}(k+1)^+ \end{align*} $$

be the canonical inclusion map. Denote

$$ \begin{align*} \partial_k=-\widetilde{\omega}^{-1}_{k+1}\iota(k): \mathscr{P}(k)^+\to\mathscr{P}(k+1)^+ \quad \text{and}\quad \partial_k^{*}=\omega^{-1}_{k+1}\iota(k): \mathscr{P}(k)^+\to\mathscr{P}(k+1)^+. \end{align*} $$

7.5

Recall that the Hall–Littlewood symmetric functions $P_{\lambda }(\mathbf {X},\textbf {t})$ are a distinguished basis of the ring of symmetric functions $\operatorname {\mathrm {Sym}}[\mathbf {X}]$ , indexed by partitions $\lambda $ . There is a remarkable family of linear operators $\mathscr {B}_n$ , $n\geq 0$ , and $\mathscr {B}_{\infty }$ on $\operatorname {\mathrm {Sym}}[\mathbf {X}]$ , defined as follows. $\mathscr {B}_{\infty }$ is the operator of left multiplication by the elementary symmetric function $e_1[\mathbf {X}]=\mathbf {X}$ and $\mathscr {B}_0$ is the operator defined by