2.1 Partitions and symmetric functions

The (French style) diagram of a partition $\mu = (\mu _1 \ge \cdots \ge \mu _k> 0)$ is the set $\{(i,j) \in {\mathbb Z}_{+}^2 : 1\leq j\leq k,\;1\leq i \leq \mu _{j}\}$ . We identify $(i,j)$ with the lattice square or box whose northeast corner has coordinates $(x,y) = (i,j)$ and refer to this box as being in column i and row j. We set $|\mu | = \mu _{1}+\cdots +\mu _{k}$ and let $\ell (\mu )=k$ be the number of nonzero parts of $\mu $ . We write $\mu ^*$ for the transpose of $\mu $ . The arm and leg of a box $b\in \mu $ are the number of boxes in $\mu $ strictly east of b and strictly north of b, respectively.

Let $\Lambda = \Lambda (X)$ be the algebra of symmetric functions in infinitely many variables $X = x_{1},x_{2},\ldots $ , with coefficients in the field ${\mathbf k} = {\mathbb Q} (q,t)$ . We follow Macdonald’s notation [Reference Macdonald and Zelevinsky29] for the graded bases of $\Lambda $ , and for the automorphism $\omega \colon \Lambda \rightarrow \Lambda $ given on Schur functions by $\omega s_{\lambda } = s_{\lambda ^*}$ . We also work with series and symmetric functions in finitely many variables ${\mathbf z} = z_1,\dots ,z_l$ . If $f(X)\in \Lambda $ is a formal symmetric function, then $f({\mathbf z} )$ or $f(z_1, \ldots , z_l)$ denotes its specialization with $X = z_1, \ldots , z_l,0,0,\ldots $ .

Given a symmetric function $f\in \Lambda $ and any expression A involving indeterminates, the plethystic evaluation $f[A]$ is defined by writing f as a polynomial in the power-sums $p_{k}$ and evaluating with $p_{k}\mapsto p_{k}[A]$ , where $p_{k}[A]$ is the result of substituting $a^{k}$ for every indeterminate a occurring in A. The variables $q, t$ from our ground field ${\mathbf k} $ count as indeterminates.

By convention, the name of an alphabet $X = x_{1},x_{2},\ldots $ stands for $x_{1}+x_{2}+\cdots $ inside a plethystic evaluation. Then $f[X] = f[x_{1}+x_{2}+\cdots ] = f(x_{1},x_{2},\ldots ) = f(X)$ . For example, the evaluation $f[X/(1-t^{-1})]$ is the image of $f(X)$ under the ${\mathbf k} $ -algebra automorphism of $\Lambda $ that sends $p_{k}$ to $p_{k}/(1-t^{-k})$ .

The modified Macdonald polynomials $\tilde {H}_\mu = \tilde {H} _{\mu }(X;q,t)$ of [Reference Garsia and Haiman17] are defined in terms of the Macdonald polynomials $Q_{\mu }(X;q,t)$ [Reference Macdonald and Zelevinsky29, VI (4.12)] or their integral forms $J_{\mu }(X;q,t)$ [Reference Macdonald and Zelevinsky29, VI (8.3)] by

(3) $$ \begin{align} \tilde{H} _{\mu }(X;q,t) = t^{\mathsf{n}(\mu )} J_{\mu}\Big[\frac{X}{1-t^{-1}};q,t^{-1} \Big] = t^{\mathsf{n}(\mu)} \Bigl( \prod_{b \in \mu}(1-q^{{\mathrm{arm}}(b)+1}t^{-{\mathrm{leg}}(b)}) \Bigr) Q_{\mu}\Big[\frac{X}{1-t^{-1}};q,t^{-1}\Big], \end{align} $$

where

(4) $$ \begin{align}\mathsf{n}(\mu ) = \sum _{i} (i-1)\mu_{i}\,.\end{align} $$

The $\tilde H_\mu (X;q,t)$ also have a direct combinatorial description [Reference Haglund, Haiman and Loehr21], which we will recall in Theorem 4.2.2.

When $q=0$ , the modified Macdonald polynomials reduce to the modified Hall-Littlewood polynomials

(5) $$ \begin{align} \tilde{H}_\mu(X;0,t) = t^{\mathsf{n}(\mu )} Q_{\mu}\Big[\frac{X}{1-t^{-1}};t^{-1}\Big], \end{align} $$

where the Hall-Littlewood polynomials $Q_{\mu }(X;t)$ are as defined in [Reference Macdonald and Zelevinsky29, III (2.11)]. At $t=1$ and $t = \infty $ , the $\tilde {H}_\mu (X;0,t)$ specialize to the complete homogeneous symmetric functions $\tilde {H}_\mu (X;0,1) = h_\mu (X)$ and Schur functions $t^{-\mathsf {n}(\mu )}\tilde {H}_\mu (X;0,t)|_{t=\infty }= s_\mu (X)$ . We will also work with the following variant of the modified Hall-Littlewood polynomials:

(6) $$ \begin{align} H_\mu(X;t) \overset{\mathit{def}}{=} t^{\mathsf{n}(\mu)}\tilde{H}_\mu(X;0,t^{-1}) = Q_{\mu}[X/(1-t)]. \end{align} $$

2.2 Weyl symmetrization and related operators

The Weyl symmetrization operator ${\boldsymbol \sigma }$ for $\operatorname {\mathrm {GL}} _{l}$ is defined by

(7) $$ \begin{align} {\boldsymbol \sigma } (f(z_1,\dots,z_l)) = \sum _{w\in \mathcal{S}_{l}} w \bigg( \frac{f(z_1,\dots,z_l)}{\prod _{i < j} (1 - z_j/z_i)} \bigg) = \sum _{w\in \mathcal{S}_{l}} w \bigg( \frac{f(z_1,\dots,z_l)}{\prod _{\alpha_{ij} \in R_{+}} (1 - z_j/z_i)} \bigg) \,, \end{align} $$

where $f \in {\mathbf k}[z_1^{\pm 1},\dots , z_l^{\pm 1}]$ is a Laurent polynomial, $\mathcal {S}_l$ acts by permuting the variables $z_1, \dots , z_l$ , and $R_+= R_+(\operatorname {\mathrm {GL}}_l) = \{\alpha _{ij} : 1\leq i < j\leq l \}$ denotes the set of positive roots for $\operatorname {\mathrm {GL}} _{l}$ , with $\alpha _{ij} = \epsilon _i -\epsilon _j \in {\mathbb Z}^l$ .

When ${\mathbf z}^\nu = z_1^{\nu _1} \cdots z_l^{\nu _l}$ for a dominant weight $\nu $ (a weight $\nu \in {\mathbb Z}^l$ is dominant if $\nu _1 \ge \cdots \ge \nu _l$ ), ${\boldsymbol \sigma } ({\mathbf z} ^{\nu }) = \chi _{\nu }$ is an irreducible $\operatorname {\mathrm {GL}} _{l}$ character. For an arbitrary weight $\gamma \in {\mathbb Z} ^{l}$ , either ${\boldsymbol \sigma } ({\mathbf z} ^{\gamma }) = \pm \chi _{\nu }$ for a suitable dominant weight $\nu $ , or ${\boldsymbol \sigma } ({\mathbf z} ^{\gamma }) = 0$ . We extend ${\boldsymbol \sigma }$ to an operator on formal ${\mathbf k}$ -linear combinations $\sum _{\gamma \in {\mathbb Z}^l} c_\gamma {\mathbf z}^\gamma $ by applying it term by term, giving an infinite formal linear combination of irreducible $\operatorname {\mathrm {GL}}_l$ characters $\sum _{\nu } a_\nu \chi _\nu = \sum _{\gamma \in {\mathbb Z}^l} c_\gamma {\boldsymbol \sigma }({\mathbf z}^\gamma ) $ . This makes sense because for each dominant weight $\nu $ , the set of monomials ${\mathbf z} ^{\gamma }$ such that ${\boldsymbol \sigma } ({\mathbf z} ^{\gamma }) = \pm \chi _{\nu }$ is finite.

Recall that the polynomial characters of $\operatorname {\mathrm {GL}} _l$ are the irreducible characters $\chi _{\nu }$ for which $\nu $ is a partition, that is, $\nu _l\geq 0$ . Given any formal ${\mathbf k}$ -linear combination $\sum _{\nu } a_{\nu } \chi _{\nu }$ of irreducible $\operatorname {\mathrm {GL}} _{l}$ characters, we define its polynomial truncation by

(8) $$ \begin{align} \operatorname{\mathrm{pol}}_X \! \Big(\sum _{\nu}a_{\nu }\hspace{.3mm} \chi _{\nu }\Big) = \sum _{ \nu_l \ge 0 }a_{\nu }\hspace{.3mm} s_{\nu }(X). \end{align} $$

In principle, the right-hand side is an infinite formal sum of symmetric functions, but, for instance, if $\sum a_{\nu } \chi _{\nu }$ is homogeneous of degree d, then the right-hand side is an ordinary symmetric function, homogeneous of degree d.

We define a related operator $\mathbf {h}_X$ on Laurent polynomials $f({\mathbf z} )$ by

(9) $$ \begin{align} \mathbf{h}_X(f({\mathbf z})) = \operatorname{\mathrm{pol}}_X {\boldsymbol \sigma } \bigg(\frac{f({\mathbf z})}{\prod_{\alpha_{ij} \in R_+} (1-z_i/z_j)} \bigg), \end{align} $$

where the factors $(1- z_{i}/z_{j})^{-1}$ are expanded as geometric series in $z_i/z_j$ before applying ${\boldsymbol \sigma }$ . When f is a monomial, it is well known [Reference Thomas39] that

(10) $$ \begin{align} \mathbf{h}_X({\mathbf z}^\gamma) &= h_\gamma(X), \end{align} $$

where for any integer vector $\gamma \in {\mathbb Z}^l$ , we define $h_\gamma = h_{\gamma _1}\cdots h_{\gamma _l}$ to be the product of complete homogeneous symmetric functions, with $h_d$ for $d\leq 0$ interpreted as $h_0 =1$ , or $h_d = 0$ for $d< 0$ .

We again extend the definition to formal linear combinations of monomials, so that $\mathbf {h}_X(\sum _{\gamma \in {\mathbb Z}^l} c_\gamma {\mathbf z}^\gamma ) = \sum _{\gamma \in {\mathbb Z}^l} c_\gamma h_\gamma (X)$ . With this interpretation, (9) still remains valid when f is a power series in $z_i/z_j$ for $i<j$ . As with $\operatorname {\mathrm {pol}}_X$ , in principle, $\mathbf {h}_X(\sum _{\gamma } c_\gamma {\mathbf z}^\gamma )$ is an infinite formal sum of symmetric functions, but for instance, if $\sum _{\gamma } c_\gamma {\mathbf z}^\gamma $ is homogeneous of degree d, then $\mathbf {h}_X(\sum _{\gamma } c_\gamma {\mathbf z}^\gamma )$ is an ordinary symmetric function, homogeneous of degree d.

2.3 Raising operator formulas for modified Hall-Littlewood polynomials

To set the stage for our raising operator formula for modified Macdonald polynomials, we review two different raising operator formulas for the modified Hall-Littlewood polynomials. Both formulas naturally reflect the geometry of the flag variety $G/B$ ; one realizes $\tilde {H}_\mu (X;0,t)$ as the graded Euler characteristic of the cotangent bundle of $G/B$ twisted by a line bundle of weight $-\mu $ , while the other is the graded Euler characteristic of the cotangent bundle of $G/P_\mu $ , where $P_\mu $ is the parabolic subgroup whose block sizes are the parts of $\mu $ . See [Reference Broer11] and [Reference Shimozono and Weyman36] for details.

The first raising operator formula for $\tilde {H}_\mu (X;0,t)$ is the one mentioned in the introduction, which we reproduce here (see [Reference Macdonald and Zelevinsky29, III (6.3)] or [Reference Milne31, (4.28) and §2]):

(11) $$ \begin{align} t^{\mathsf{n}(\mu)}\tilde{H}_\mu(X;0,t^{-1}) = H_\mu(X;t) = \operatorname{\mathrm{pol}}_X {\boldsymbol \sigma }\Big(\frac{{\mathbf z}^\mu}{\prod_{\alpha_{ij} \in R_+}(1 - t \hspace{.3mm} z_i/z_j)}\Big)\,, \end{align} $$

where the denominator factors are expanded as geometric series in accordance with Remark 2.2.1.

A second raising operator formula follows from the work of Weyman and Shimozono-Weyman (see [Reference Weyman40, Theorem 6.10] and [Reference Shimozono and Weyman36, §2.3 (2) and (2.3)–(2.5)]). In this formula, the input partition $\mu $ appears in the set of roots, instead of in the weight ${\mathbf z}^\mu $ , as it does in formula (11). Given a partition $\mu $ of l, consider the set partition of $\{1,\ldots ,l \}$ into intervals of lengths $\mu _{\ell (\mu )}, \dots , \mu _1$ , and let $B_\mu $ denote the set of roots $\alpha _{ij}$ such that $i < j$ appear in distinct blocks of this partition. Then

(12) $$ \begin{align} \tilde{H}_\mu(X;0,t) = \omega \operatorname{\mathrm{pol}}_X {\boldsymbol \sigma }\Big(\frac{z_1\cdots z_l}{\prod_{\alpha_{ij} \in B_\mu}(1 - t \hspace{.3mm} z_i/z_j)}\Big). \end{align} $$

Here, we chose to take the parts of $\mu $ in reverse order for compatibility with the formula (21) given later, but the order does not actually matter in (12).

Remark 2.3.1. Formulas such as (11) and (12) are traditionally written using an informal notation – as in [Reference Macdonald and Zelevinsky29], [Reference Milne31, (4.28)], or [Reference Tamvakis38, §2] – in which formula (11), for example, would be written as

(13) $$ \begin{align} t^{\mathsf{n}(\mu)}\tilde{H}_\mu(X;0,t^{-1}) = \frac{ 1 } { \prod_{\alpha_{i j} \in R_+} \big(1-t \hspace{.3mm} \mathbf{R}_{ij}\big)} \cdot s_{\mu}\,, \end{align} $$

with raising ‘operators’ $\mathbf {R}_{ij}$ which act on the subscript of a Schur function $s_\gamma $ by $\mathbf {R}_{ij} \gamma = \gamma + \epsilon _i - \epsilon _j$ . Here, Schur functions indexed by non-partition weights are defined by $s_\gamma (X) = \operatorname {\mathrm {pol}} _{X}{\boldsymbol \sigma }({\mathbf z}^\gamma )$ , which is equal to 0 or to $\pm 1$ times a Schur function of partition weight. Note that all the raising operators must be applied before converting Schur functions of non-partition weights to ones indexed by partition weights. The $\mathbf {R}_{ij}$ are not true operators (e.g., $\mathbf {R}_{23}s_{(1,1,1)} = \operatorname {\mathrm {pol}} _{X}{\boldsymbol \sigma }(z_1z_2^2)= 0$ but $\mathbf {R}_{12}\mathbf {R}_{23}s_{(1,1,1)} = s_{(2,1,0)}\not =0$ ), so we think of (13) as a convenient but informal version of (11).

3.1 The formula

We in fact give many different raising operator formulas for $ \tilde {H}_\mu (X;q,t)$ , one for each rearrangement $\beta $ of the parts of $\mu ^*$ .

For $\beta = (\beta _1, \dots , \beta _k) \in {\mathbb Z}_{+}^k$ , we define the column diagram of $\beta $ to be the set

(14) $$ \begin{align} \boldsymbol {\beta} = \big\{(i,j) \in {\mathbb Z}_{+}^2 : \hspace{.3mm} 1 \le i \le k, \, 1 \le j \le \beta_i \big\}. \end{align} $$

We identify $(i,j)$ with the box whose northeast corner has coordinates $(x,y) = (i,j)$ ; we say that this box is in column i and row j. The reading order on $\boldsymbol {\beta }$ is the total order $\prec $ on the boxes of $\boldsymbol {\beta }$ given by $(i,j) \prec (i',j')$ if $j> j'$ , or $j = j'$ and $i < i'$ . We let ${\boldsymbol {\beta }[1]},\dots , {\boldsymbol {\beta }[l]}$ denote the boxes of $\boldsymbol {\beta }$ listed in increasing reading order, which is the list of boxes of $\boldsymbol {\beta }$ read by rows from left to right starting from the top row, as shown in Example 3.1.5. For a box $b=(i,j)$ , ${\mathrm {south}}(b)=(i,j-1)$ denotes the box immediately south of b. Define subsets of $R_+ = R_+(\operatorname {\mathrm {GL}}_l)$ by

(15) $$ \begin{align} R_{\beta} = \big\{ \alpha_{ij} \in R_+ : \hspace{.3mm} {\mathrm{south}}({\boldsymbol {\beta}[i]}) \in \boldsymbol {\beta}, \, {\mathrm{south}}({\boldsymbol {\beta}[i]}) \hspace{.3mm} \preceq \hspace{.3mm} {\boldsymbol {\beta}[j]} \big\}, \end{align} $$

(16) $$ \begin{align} \widehat{R}_{\beta} = \big\{ \alpha_{ij} \in R_+ : \hspace{.3mm} {\mathrm{south}}({\boldsymbol {\beta}[i]}) \in \boldsymbol {\beta}, \, {\mathrm{south}}({\boldsymbol {\beta}[i]}) \hspace{.3mm} \prec \hspace{.3mm} {\boldsymbol {\beta}[j]} \big\}. \end{align} $$

For $\beta \in {\mathbb Z}_{+}^k$ and a box $b = (i,j) \in \boldsymbol {\beta }$ with $j> 1$ , define the arm and leg of b by

(17) $$ \begin{align} {\mathrm{leg}}(b) &= \beta_i -j = \text{(number of boxes strictly north of }b), \end{align} $$

(18) $$ \begin{align} {\mathrm{arm}}(b) &= \big| \big\{i' \in \{1,\dots, i-1\} : j\!-\!1 \le \beta_{i'} < \beta_i \big\} \sqcup \! \big\{i' \in \{i+1,\dots, k\} : j \le \beta_{i'} \le \beta_i \big\} \big|.\!\!\!\!\! \end{align} $$

In words, ${\mathrm {arm}} (b)$ is the number of boxes strictly east of b in columns of height $\beta _{i'}\leq \beta _{i}$ or strictly west of ${\mathrm {south}} (b)$ in columns of height $\beta _{i'}<\beta _{i}$ .

Example 3.1.1. For $\beta = (3,2,4,3,4,2,1,3)$ , the column diagram of $\beta $ is displayed below, along with the ${\mathrm {arm}}$ of box $\bullet =(4,2)$ where the a’s mark the boxes contributing to ${\mathrm {arm}}(\bullet )$ .

(19)

Definition 3.1.3. For $\beta \in {\mathbb Z}_{+}^k$ , define the Macdonald series by

(20) $$ \begin{align} \mathbf{H}_{\beta}({\mathbf z};q,t) = {\boldsymbol \sigma } \Bigg( \frac{ \prod_{\alpha_{ij} \in R_{\beta} \setminus \widehat{R}_{\beta}} \big(1- q^{{\mathrm{arm}}({\boldsymbol {\beta}[i]})+1}\hspace{.3mm} t^{-{\mathrm{leg}}({\boldsymbol {\beta}[i]})} z_i/z_j \big) \prod_{\alpha_{ij} \in \widehat{R}_{\beta}} \big(1-q\hspace{.3mm} t\, z_i/z_j \big)} {\prod_{\alpha_{ij} \in R_+} \big(1-q \, z_i/z_j\big) \prod_{\alpha_{ij} \in R_{\beta}} \big(1-t \, z_i/z_j\big)}\Bigg), \end{align} $$

which we interpret as an infinite formal linear combination of irreducible $\operatorname {\mathrm {GL}} _{l}$ characters by expanding the denominator factors as geometric series, in accordance with Remark 2.2.1.

We have the following raising operator formulas for the modified Macdonald polynomials $ \tilde {H}_{\mu }(X; q,t)$ . The proof will be given in §4.

Theorem 3.1.4. For any partition $\mu $ of l and any rearrangement $\beta $ of $\mu ^*$ ,

(21) $$ \begin{align} \tilde{H}_\mu(X;q,t) = \omega \operatorname{\mathrm{pol}} _{X}\! \big(z_1\cdots z_l \, \mathbf{H}_{\beta}({\mathbf z};q, t) \big), \end{align} $$

where $\operatorname {\mathrm {pol}}_X$ is as defined in (8).

The case of Theorem 3.1.4 when $\beta = \mu ^*$ is the formula (2) previewed in the introduction.

Example 3.1.5. (i) For $\beta = (4,2)$ , $l = 6$ , we visualize the data for the series $\mathbf {H}_{\beta }$ below, with the subsets of roots $R_\beta $ and $\widehat {R}_\beta $ drawn in an $l \times l$ grid, labeled by matrix-style coordinates and specified by the legend, and with large circles marking the presence of the factors involving arm and leg, which are $(1-q \hspace {.3mm} z_1/z_2)$ , $(1-qt^{-1}z_2/z_3)$ , $(1-q^2t^{-2}z_3/z_5)$ , $(1-q \hspace {.3mm} z_4/z_6)$ .

(ii) For $\beta = (1,4,2,4)$ , $l = 11$ , we visualize the data for $\mathbf {H}_{\beta }$ with the same conventions; the factors involving arm and leg, marked by the large circles, are $(1-q^2 \hspace {.3mm} z_1/z_3)$ , $(1-qz_2/z_4)$ , $(1-q^2t^{-1}z_3/z_5)$ , $(1-q^2t^{-1} \hspace {.3mm} z_4/z_7)$ , $(1-q^4t^{-2} \hspace {.3mm} z_5/z_9)$ , $(1-q^2 \hspace {.3mm} z_6/z_{10})$ , $(1-q^3t^{-2} \hspace {.3mm} z_7/z_{11})$ .

3.2 Specializations

It is instructive to see how the well-known specializations of Macdonald polynomials, $\tilde {H} _{\mu }(X;1,1)=e_{1}^{|\mu |}(X)$ and the Hall-Littlewood specialization $\tilde {H} _{\mu }(X;0,t)$ , can be recovered from formula (21). This can be done for general $\beta $ , but it is a little simpler when $\beta $ is a partition. Accordingly, for this subsection, we now consider only the case $\beta = \mu ^{*}$ .

First, we consider the specialization $q=t=1$ . After specializing, the arm-leg and $(1-q\,t \,z_i/z_j)$ numerator factors in the definition (20) of $\mathbf {H}_\beta $ cancel with the $(1-t\,z_i/z_j)$ factors in the denominator. Hence,

(22) $$ \begin{align} \mathbf{H}_\beta({\mathbf z};1,1) = {\boldsymbol \sigma }\Big(\frac{1}{\prod_{\alpha_{ij} \in R_+}(1 - z_i/z_j)}\Big). \end{align} $$

Then, using (9) and (10), we recover the specialization

(23) $$ \begin{align} \tilde{H}_\mu(X;1,1) = \omega \operatorname{\mathrm{pol}} _{X}(z_1\cdots z_l \hspace{.3mm} \mathbf{H}_\beta({\mathbf z};1,1)) = e_{1^l} = e_1^l. \end{align} $$

Now consider the specialization $q=0$ . After specializing (20), all numerator factors and the $(1-q z_i/z_j)$ denominator factors reduce to 1. This gives

(24) $$ \begin{align} z_1\cdots z_l \, \mathbf{H}_\beta({\mathbf z};0,t) = {\boldsymbol \sigma }\Big(\frac{z_1\cdots z_l}{\prod_{\alpha_{ij} \in R_\beta}(1 - t \hspace{.3mm} z_i/z_j)}\Big). \end{align} $$

Let $B_\mu $ be the set of roots defined before (12), which can also be described as the set of positive roots above a block diagonal matrix with block sizes $\mu _{\ell (\mu )}, \dots , \mu _1$ . Now $R_\beta $ is contained in $B_\mu $ and $B_\mu \setminus R_\beta $ consists of triangular regions of roots between each pair of consecutive blocks. We reach the Hall-Littlewood raising operator formula using the identity

(25) $$ \begin{align} {\boldsymbol \sigma }\Big(\frac{z_1\cdots z_l}{\prod_{\alpha_{ij} \in B_\mu}(1 - t \hspace{.3mm} z_i/z_j)}\Big) = {\boldsymbol \sigma }\Big(\frac{z_1\cdots z_l}{\prod_{\alpha_{ij} \in R_\beta}(1 - t \hspace{.3mm} z_i/z_j)}\Big). \end{align} $$

This is proven by removing these triangular regions from $B_\mu $ one root at a time (starting with the bottommost region), and using the following simplified version of [Reference Blasiak, Morse, Pun and Summers9, Lemma 8.9] to show that the corresponding functions remain the same at each step.

Lemma 3.2.1. Let $k \in \{1,\dots , l-1\}$ and suppose that $B \subseteq R_+(\operatorname {\mathrm {GL}}_l)$ is a set of roots such that $\prod _{\alpha _{ij}\in B} (1-tz_i/z_j)$ is fixed by the simple reflection $s_k$ . If B contains a root $\alpha = \alpha _{k+1,j}$ for some $j> k+1$ , then

(26) $$ \begin{align} {\boldsymbol \sigma }\Big(\frac{z_1\cdots z_l}{\prod_{\alpha_{ij} \in B}(1 - t \hspace{.3mm} z_i/z_j)}\Big) = {\boldsymbol \sigma }\Big(\frac{z_1\cdots z_l}{\prod_{\alpha_{ij} \in B \setminus \alpha}(1 - t \hspace{.3mm} z_i/z_j)}\Big). \end{align} $$

Combining (24) and (25) now shows that the raising operator formula (21) for modified Macdonald polynomials reduces at $q=0$ to the raising operator formula (12) for Hall-Littlewood polynomials.

4.1 LLT polynomials

We recall the definition and basic properties of LLT polynomials [Reference Lascoux, Leclerc and Thibon26], using the ‘attacking inversions’ formulation from [Reference Haglund, Haiman, Loehr, Remmel and Ulyanov23].

A skew diagram is a difference $\nu = \lambda / \mu $ of partition diagrams $\mu \subseteq \lambda $ . The content of a box $b = (i,j)$ in row j, column i of a (skew) diagram is $c(b) = i-j$ .

Let ${\boldsymbol \nu } = (\nu _{(1)},\ldots ,\nu _{(k)})$ be a tuple of skew diagrams. We consider the set of boxes in ${\boldsymbol \nu } $ to be the disjoint union of the sets of boxes in the $\nu _{(i)}$ , and define the adjusted content of a box $a\in \nu _{(i)}$ to be $\tilde {c} (a) = c(a)+i\epsilon $ , where $\epsilon $ is a fixed positive number such that $k \epsilon <1$ .

A diagonal in ${\boldsymbol \nu } $ is the set of boxes of a fixed adjusted content – that is, a diagonal of fixed content in one of the $\nu _{(i)}$ .

The reading order on ${\boldsymbol \nu } $ is the total ordering $<$ on the boxes of ${\boldsymbol \nu } $ such that $a<b \Rightarrow \tilde {c} (a)\leq \tilde {c}(b)$ and boxes on each diagonal increase from southwest to northeast. See Example 4.2.3. An attacking pair is an ordered pair of boxes $(a,b)$ in $ {\boldsymbol \nu }$ such that $a<b$ in reading order and $0<\tilde {c}(b)-\tilde {c}(a)<1$ .

A semistandard tableau on the tuple ${\boldsymbol \nu } $ is a map $T\colon {\boldsymbol \nu } \rightarrow {\mathbb Z} _{+}$ which restricts to a semistandard Young tableau on each component $\nu _{(i)}$ . The set of these is denoted $\operatorname {\mathrm {SSYT}} ({\boldsymbol \nu } )$ . An attacking inversion in T is an attacking pair $(a,b)$ such that $T(a)>T(b)$ . The number of attacking inversions in T is denoted $\operatorname {\mathrm {inv}} (T)$ .

Definition 4.1.1. The LLT polynomial indexed by a tuple of skew diagrams ${\boldsymbol \nu } $ is the generating function

(27) $$ \begin{align} {\mathcal G}_{{\boldsymbol \nu } }(X;q) = \sum _{T\in \operatorname{\mathrm{SSYT}} ({\boldsymbol \nu })}q^{\operatorname{\mathrm{inv}} (T)}{\mathbf x} ^{T}, \end{align} $$

where ${\mathbf x} ^{T} = \prod _{a\in {\boldsymbol \nu } } x_{T(a)}$ . By [Reference Haglund, Haiman, Loehr, Remmel and Ulyanov23, Reference Lascoux, Leclerc and Thibon26], ${\mathcal G}_{{\boldsymbol \nu } }(X;q) $ is known to be symmetric.

4.2 The Haglund-Haiman-Loehr formula

Haglund-Haiman-Loehr [Reference Haglund, Haiman and Loehr21] gave a formula for the modified Macdonald polynomials $H_\mu (X;q,t)$ as a positive sum of LLT polynomials, and this was generalized in [Reference Haglund, Haiman and Loehr22] to give many different expressions for $H_\mu (X;q,t)$ as a positive sum of LLT polynomials, one for each rearrangement $\beta $ of $\mu ^*$ . We now recall this formula.

A ribbon is a connected skew shape containing no $2 \times 2$ block of boxes.

For $\beta \in {\mathbb Z}_{+}^k$ , let $V_\beta = \{({\boldsymbol {\beta }[i]},{\boldsymbol {\beta }[j]}) : {\boldsymbol {\beta }[j]} = {\mathrm {south}}({\boldsymbol {\beta }[i]}) \}$ be the set of ordered pairs of boxes that form vertical dominoes in $\boldsymbol {\beta }$ .

Theorem 4.2.2 [Reference Haglund, Haiman and Loehr21, Reference Haglund, Haiman and Loehr22].

Let $\mu $ be a partition and let $\beta $ be any rearrangement of $\mu ^{*}$ . Then

(28) $$ \begin{align} \tilde{H}_\mu(X;q,t) = \sum_{S \subseteq V_\beta} \Big( \prod_{({\boldsymbol {\beta}[i]},{\boldsymbol {\beta}[j]}) \in S} q^{-{\mathrm{arm}}({\boldsymbol {\beta}[i]})} \hspace{.3mm} t^{{\mathrm{leg}}({\boldsymbol {\beta}[i]})+1} \Big) {\mathcal G}_{{\boldsymbol \nu }(S)}(X;q). \end{align} $$

Theorem 4.2.2 in the case that $\beta $ is a partition is immediate from [Reference Haglund, Haiman and Loehr21, Theorem 2.2, equation (23), and Proposition 3.4], while the generalization to any composition $\beta $ is addressed in [Reference Haglund, Haiman and Loehr22, Theorem 5.1.1]. Note that our conventions for diagrams, arms, and legs are the same as those in [Reference Haglund, Haiman and Loehr22] except that we have reversed the order of the columns (which makes our notation consistent with that in [Reference Haglund, Haiman and Loehr21]); we have also used the symmetry property $H_\mu (X;q,t) = H_{\mu ^*}(X;t,q)$ to translate from the exact version stated in [Reference Haglund, Haiman and Loehr22, Theorem 5.1.1].

Example 4.2.3. For $\beta = (2,3)$ , the 8 summands appearing on the right-hand side of (28) are illustrated by drawing ${\boldsymbol \nu }(S)$ with boxes labeled in reading order and with the corresponding coefficient $\prod _{({\boldsymbol {\beta }[i]},{\boldsymbol {\beta }[j]}) \in S} q^{-{\mathrm {arm}}({\boldsymbol {\beta }[i]})} \hspace {.3mm} t^{{\mathrm {leg}}({\boldsymbol {\beta }[i]})+1}$ beside it; the vertical dominoes of β are denoted v ₁ = ( β [1], β [3]), v ₂ = ( β [2], β [4]), v ₃ = ( β [3], β [5]). The arm and leg statistics for $\boldsymbol {\beta }$ are shown on the left.

4.3 A formula for $\nabla $ on any LLT polynomial

The operator $\nabla $ , introduced in [Reference Bergeron, Garsia, Haiman and Tesler2], is the linear operator on symmetric functions which acts diagonally on the basis of modified Macdonald polynomials $\tilde {H}_{\mu }(X;q,t)$ by $\nabla \tilde {H} _{\mu } = q^{\mathsf {n}(\mu ^{*})} t^{\mathsf {n}(\mu )}\tilde {H} _{\mu }$ .

In [Reference Blasiak, Haiman, Morse, Pun and Seelinger7], we give a raising operator formula for $\nabla $ on any LLT polynomial. The formula takes a simpler form in the case that the LLT polynomial is indexed by a tuple of ribbons. We state the result for the tuple ${\boldsymbol \nu }(S)$ in Definition 4.2.1, making use of the notation $V_\beta , R_{\beta }, \widehat {R}_{\beta }$ defined in §4.2, (15), and (16). Also let $A_\beta $ denote the number of attacking pairs in ${\boldsymbol \nu }(S)$ , which depends only on $\beta $ and not on $S \subseteq V_\beta $ .

Theorem 4.3.1 [Reference Blasiak, Haiman, Morse, Pun and Seelinger7].

For $\beta \in {\mathbb Z}_{+}^k$ and $S \subseteq V_\beta $ , consider the tuple of ribbons ${\boldsymbol \nu }(S)$ . We have the following formula for the operator $\nabla $ applied to the LLT polynomial ${\mathcal G}_{{\boldsymbol \nu }(S)}(X;q)$ :

(29) $$ \begin{align} &\nabla {\mathcal G}_{{\boldsymbol \nu }(S)}(X;q) = \omega\operatorname{\mathrm{pol}}_X {\boldsymbol \sigma } \bigg( (-qt)^{|V_\beta\setminus S|} q^{A_\beta} \frac{z_1 \cdots z_l \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]}) \in V_\beta \setminus S} z_i/z_j \prod_{\alpha_{ij} \in \widehat{R}_{\beta}} \! \big(1-q\hspace{.3mm} t\, z_i/z_j \big)} { \prod_{\alpha_{ij} \in R_+} \! \big(1-q \, z_i/z_j\big) \prod_{\alpha_{ij} \in R_{\beta}} \! \big(1-t \, z_i/z_j\big)}\bigg), \end{align} $$

where $l= |\beta |$ and $R_+ = R_+(\operatorname {\mathrm {GL}} _l)$ .

4.4 Proof of Theorem 3.1.4

Applying $\nabla $ to (28) and substituting (29) into this yields

(30) $$ \begin{align} \nabla \tilde{H}_\mu(X;q,t) = \omega \operatorname{\mathrm{pol}}_X {\boldsymbol \sigma } \bigg(\frac{z_1 \cdots z_l \cdot \Upsilon \cdot \prod_{\alpha_{ij} \in \widehat{R}_{\beta} } \big(1-q\hspace{.3mm} t\, z_i/z_j \big)} { \prod_{\alpha_{ij} \in R_+} \big(1-q \, z_i/z_j\big) \prod_{\alpha_{ij} \in R_{\beta}} \big(1-t \, z_i/z_j\big)}\bigg), \end{align} $$

where

(31) $$ \begin{align} \Upsilon = \sum_{S \subseteq V_\beta} (-qt)^{|V_\beta \setminus S|} q^{A_\beta} \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]})\in S} q^{-{\mathrm{arm}}({\boldsymbol {\beta}[i]})} \hspace{.3mm} t^{{\mathrm{leg}}({\boldsymbol {\beta}[i]})+1} \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]})\in V_\beta \setminus S} z_i/z_j. \end{align} $$

Defining $d = \! A_\beta - \mathsf {n}(\mu ^*) - \sum _{({\boldsymbol {\beta }[i]},{\boldsymbol {\beta }[j]}) \in V_\beta } {\mathrm {arm}}({\boldsymbol {\beta }[i]})$ and using $\mathsf {n}(\mu ) = \! \sum _{({\boldsymbol {\beta }[i]}, {\boldsymbol {\beta }[j]})\in V_\beta } ({\mathrm {leg}}({\boldsymbol {\beta }[i]})+1)$ , the quantity $\Upsilon $ can be simplified as follows:

$$ \begin{align*} \Upsilon &= q^{\mathsf{n}(\mu^*)+d} \, t^{\mathsf{n}(\mu)} \sum_{S \subseteq V_\beta}\Big( (-qt)^{|V_\beta\setminus S|} \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]})\in V_\beta} \!\!\!\! q^{{\mathrm{arm}}({\boldsymbol {\beta}[i]})} \, t^{-{\mathrm{leg}}({\boldsymbol {\beta}[i]})-1} \\ & \hphantom{q^{\mathsf{n}(\mu^*)+d} \, t^{\mathsf{n}(\mu)} \sum_{S \subseteq V_\beta}\Big( (-qt)^{|V_\beta\setminus S|} \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]})\in V_\beta}} \times \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]})\in S} \!\!\!\! q^{-{\mathrm{arm}}({\boldsymbol {\beta}[i]})} \, t^{{\mathrm{leg}}({\boldsymbol {\beta}[i]})+1} \!\!\! \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]})\in V_\beta \setminus S} \!\!\!\!\!\! z_i/z_j \Big) \\ &= q^{\mathsf{n}(\mu^*)+d} \hspace{.3mm} t^{\mathsf{n}(\mu)} \sum_{S \subseteq V_\beta} \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]})\in V_\beta\setminus S} \!\! \big(-q^{{\mathrm{arm}}({\boldsymbol {\beta}[i]})+1} \, t^{-{\mathrm{leg}}({\boldsymbol {\beta}[i]})} z_i/z_j \big)\\ &= q^{\mathsf{n}(\mu^*)+d} \hspace{.3mm} t^{\mathsf{n}(\mu)} \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]}) \in V_\beta} \big(1- q^{{\mathrm{arm}}({\boldsymbol {\beta}[i]})+1}\hspace{.3mm} t^{-{\mathrm{leg}}({\boldsymbol {\beta}[i]})} z_i/z_j \big). \end{align*} $$

Thus, plugging this back in for $\Upsilon $ in (30) and recalling the definition of $\mathbf {H}_{\beta }({\mathbf z};q, t)$ (Definition 3.1.3), we have

$$\begin{align*}\nabla \tilde{H}_\mu(X;q,t) = q^{\mathsf{n}(\mu^*)+d} \hspace{.3mm} t^{\mathsf{n}(\mu)} \omega\operatorname{\mathrm{pol}}_X \! \big(z_1\cdots z_l \, \mathbf{H}_{\beta}({\mathbf z};q, t) \big). \end{align*}$$

By the definition of $\nabla $ , this implies $\tilde {H}_\mu (X;q,t) = q^{d} \omega \operatorname {\mathrm {pol}}_X \! \big (z_1\cdots z_l \, \mathbf {H}_{\beta }({\mathbf z};q, t) \big ).$ It remains to show that $d=0$ . This follows from the fact that the coefficient of $s_{1^l}(X) = \operatorname {\mathrm {pol}}_X(z_1\cdots z_l)$ in the Schur expansion of both $\omega \tilde {H}_\mu (X;q,t)$ and $\operatorname {\mathrm {pol}}_X \! \big (z_1\cdots z_l \, \mathbf {H}_{\beta }({\mathbf z};q, t) \big )$ is 1; the former is well known, while the latter can be seen directly by expanding the series $z_1\cdots z_l \, \mathbf {H}_{\beta }({\mathbf z};q, t)$ to see that it is equal to ${\boldsymbol \sigma }(z_1\cdots z_l)= \chi _{1^l}$ plus terms of the form $a_\nu \chi _\nu $ for $\nu> 1^l$ in dominance order.

5.2 The $1,n$ -Macdonald polynomials

By [Reference Blasiak, Haiman, Morse, Pun and Seelinger7, Proposition 4.5.1], $\tilde {H}_\mu ^{m,1} = \tilde {H}_\mu [-M X^{m,1}] \cdot 1 = \nabla ^m \tilde {H}_\mu = q^{m \hspace {.3mm} \mathsf {n}(\mu ^*)}t^{m \hspace {.3mm} \mathsf {n}(\mu )} \tilde {H}_\mu $ , so this case is familiar. The $1,n$ -Macdonald polynomials, however, carry essentially the same data as the Macdonald series ${\mathbf H} _{\beta }$ by the following theorem, which will be proven in §5.3 as part of a more general result (see Remark 5.3.4 (ii)).

Theorem 5.2.1. For any partition $\mu $ of l and any rearrangement $\beta $ of $\mu ^{*}$ ,

(33) $$ \begin{align} \tilde{H}_{\mu}^{1,n}(X;q,t) = \omega\operatorname{\mathrm{pol}}_X \! \Big(q^{\mathsf{n}(\mu^*)}t^{\mathsf{n}(\mu)}\, (z_1\cdots z_l)^n \, \mathbf{H}_{\beta}({\mathbf z};q, t) \Big). \end{align} $$

Hence, also $\mathbf {H}_{\beta }({\mathbf z};q, t) = q^{-\mathsf {n}(\mu ^*)}t^{-\mathsf {n}(\mu )} \lim _{n \to \infty } (z_1\cdots z_l)^{-n}(\omega \tilde {H}_{\mu }^{1,n})(z_1,\dots , z_l).$

This also shows that $\mathbf {H}_{\beta }({\mathbf z};q, t)$ depends only on the partition rearrangement $\mu ^*$ of $\beta $ .

The Macdonald polynomial $\tilde {H} _{\mu }(X;q,t)$ is known [Reference Haiman24] to be Schur positive (i.e., the coefficients $\tilde {K} _{\lambda ,\mu }(q,t)$ in its Schur expansion are in ${\mathbb N} [q,t]$ ). Based on extensive computations, we were led to the following positivity conjecture for the $1,n$ -Macdonald polynomials, which generalizes the positivity theorem for Macdonald polynomials.

Conjecture 5.2.2. The $1,n$ -Macdonald polynomials $\tilde {H}_{\mu }^{1,n}(X;q,t)$ are Schur positive.

Equivalently (by Theorem 5.2.1) for any partition $\mu $ of l and rearrangement $\beta $ of $\mu ^*$ , the series ${\mathbf H}_\beta ({\mathbf z};q,t)$ is a positive sum of irreducible $\operatorname {\mathrm {GL}}_l$ characters; that is, the coefficients in

(34) $$ \begin{align} {\mathbf H}_\beta({\mathbf z};q,t) = \sum_{\nu } {\mathbf K} _{\nu , \mu}(q,t)\, \chi_\nu \end{align} $$

are polynomials ${\mathbf K} _{\nu , \mu }(q,t)\in {\mathbb N}[q,t]$ with nonnegative integer coefficients.

Example 5.2.4. The Schur expansions of the $1,n$ -Macdonald polynomials $\tilde {H}^{1,n}_\mu (X;q,t)$ for $n=2$ and $|\mu | = 2, 3$ , written as in (33), are

$$ \begin{align*} \tilde{H}^{1,2}_{2} \, &= \omega \, q \big( q^2s_{4} + q \hspace{.3mm} s_{31} + s_{22} \big)\\ \tilde{H}^{1,2}_{11} \, &= \omega \,t \big( t^2s_{4} + t \hspace{.3mm} s_{31} + s_{22} \big) \\ \tilde{H}^{1,2}_{3} \, &= \omega \,q^3 \big( q^6s_{6} + (q^5 + q^4)s_{51} + (q^4 + q^3 + q^2)s_{42} + q^3s_{33} + q^3s_{411} + (q^2 + q)s_{321} + s_{222}\big) \\ \tilde{H}^{1,2}_{21} \, &= \omega \, qt \big( q^2t^2s_{6} + (q^2t + qt^2)s_{51} + (q^2 + qt + t^2)s_{42} + qt \hspace{.3mm} s_{33} + qt \hspace{.3mm} s_{411} + (q + t)s_{321} + s_{222} \big) \\ \tilde{H}^{1,2}_{111} \, &= \omega\, t^3 \big( t^6s_{6} + (t^5 + t^4)s_{51} + (t^4 + t^3 + t^2)s_{42} + t^3s_{33} + t^3s_{411} + (t^2 + t)s_{321} + s_{222} \big). \end{align*} $$

We obtain several specializations of the $1,n$ -Macdonald polynomials easily from the raising operator formula in Theorem 5.2.1.

Proposition 5.2.6. Let $\mu $ be a partition of l. The $q=t=1$ , $q=1$ and $q=0$ specializations of the $1,n$ -Macdonald polynomials are given by

(35) $$ \begin{align} &\tilde{H}^{1,n}_\mu(X;1,1) = e_{(n^l)}(X), \end{align} $$

(36) $$ \begin{align} &\big(q^{-\mathsf{n}(\mu^*)} t^{-\mathsf{n}(\mu)}\tilde{H}^{1,n}_\mu(X;q,t) \big)|_{q=0} = \omega \operatorname{\mathrm{pol}}_X {\boldsymbol \sigma }\Big(\frac{(z_1\cdots z_l)^n}{\prod_{\alpha_{ij} \in B_\mu} (1 - t \hspace{.3mm} z_i/z_j)}\Big), \end{align} $$

(37) $$ \begin{align} &\tilde{H}^{1,n}_\mu(X;1,t) = t^{\mathsf{n}(\mu)} \prod_{r = 1}^{\mu_1}\omega H_{(n^{\nu_r})}(X;t), \end{align} $$

where $\nu = \mu ^*$ , $H_\mu (X;t) = t^{\mathsf {n}(\mu )}\tilde {H}_\mu (X;0,t^{-1})$ is as in (6), and $B_\mu $ is as defined before (12).

Proof. Throughout this proof, we only need the case $\beta = \mu ^* = \nu $ of Theorem 5.2.1.

By the same argument as in §3.2, the $q=t=1$ specialization of $\operatorname {\mathrm {pol}}_X ((z_1\cdots z_l)^n\mathbf {H}_\nu )$ is the complete homogeneous symmetric function $h_{(n^l)}$ . Hence, (35) follows from Theorem 5.2.1. Similarly, (36) follows from Theorem 5.2.1 and from noting that $\operatorname {\mathrm {pol}}_X ((z_1\cdots z_l)^n\mathbf {H}_\nu ({\mathbf z}; 0,t))$ can be simplified just as $\operatorname {\mathrm {pol}}_X (z_1\cdots z_l \, \mathbf {H}_\nu ({\mathbf z}; 0,t))$ was in §3.2.

We now prove (37). By Theorem 5.2.1 and (9),

(38) $$ \begin{align} q^{-\mathsf{n}(\mu^*)}t^{-\mathsf{n}(\mu)} \omega\hspace{.3mm} \tilde{H}^{1,n}_{\mu }(X;q,t) = \mathbf{h}_X\Big( \hspace{.3mm} (z_1\cdots z_l)^n \prod_{\alpha_{ij} \in R_+}(1-z_i/z_j)\widehat{\mathbf{H}}_{\nu}\Big), \end{align} $$

where $\widehat {\mathbf {H}}_\nu ({\mathbf z};q,t)$ is the expression inside the ${\boldsymbol \sigma }(\cdot )$ on the right side of (20), with $\beta $ equal to $\nu $ (so that $\mathbf {H}_\nu = {\boldsymbol \sigma }(\widehat {\mathbf {H}}_\nu )$ ). Let $C_1, \dots , C_{\mu _1}$ denote the columns of $\boldsymbol {\nu }$ and note that $R_\nu \setminus \widehat {R}_\nu $ is equal to $\bigsqcup _{r = 1}^{\mu _1} \{\alpha _{ij} : {\boldsymbol {\nu }[j]}={\mathrm {south}}({\boldsymbol {\nu }[i]}), \, {\boldsymbol {\nu }[i]} \in C_r \}$ . Hence, setting $q=1$ in (38) yields

(39) $$ \begin{align} t^{-\mathsf{n}(\mu)} \omega \tilde{H}^{1,n}_{\mu }(X;1,t) & = \mathbf{h}_X\bigg( \prod_{r = 1}^{\mu_1} \frac{\big(\prod_{{\boldsymbol {\nu}[i]} \in C_r} z_i^n\big) \prod_{\alpha_{ij} \in R_\nu \setminus \widehat{R}_\nu, \, {\boldsymbol {\nu}[i]} \in C_r } \big(1- t^{-{\mathrm{leg}}({\boldsymbol {\nu}[i]})} z_i/z_j \big)} {\prod_{\alpha_{ij} \in R_\nu \setminus \widehat{R}_\nu, \, {\boldsymbol {\nu}[i]} \in C_r}\big(1-t z_i/z_j \big)} \bigg) \end{align} $$

(40) $$ \begin{align} &\qquad\qquad\qquad\qquad\,\, = \prod_{r = 1}^{\mu_1}\mathbf{h}_X\bigg( \frac{\big(\prod_{{\boldsymbol {\nu}[i]} \in C_r} z_i^n\big) \prod_{\alpha_{ij} \in R_\nu \setminus \widehat{R}_\nu, \, {\boldsymbol {\nu}[i]} \in C_r } \big(1- t^{-{\mathrm{leg}}({\boldsymbol {\nu}[i]})} z_i/z_j \big)} {\prod_{\alpha_{ij} \in R_\nu \setminus \widehat{R}_\nu\, {\boldsymbol {\nu}[i]} \in C_r}\big(1-t z_i/z_j \big)} \bigg), \end{align} $$

where the second equality follows from (10) and the fact that $h_\gamma = h_\delta $ for $\gamma , \delta \in {\mathbb Z}^l$ which are rearrangements of each other.

The factor $\mathbf {h}_X(\cdot )$ in (40) for a given index r is equal to $t^{-\mathsf {n}(1^{\nu _r})} \omega \tilde {H}^{1,n}_{(1^{\nu _r})}(X;1,t)$ , by the computation we have just done, but with the partition $1^{\nu _r}$ in place of $\mu $ . It follows that

(41) $$ \begin{align} \tilde{H}^{1,n}_{\mu }(X;1,t) = \prod_{r = 1}^{\mu_1}\tilde{H}^{1,n}_{(1^{\nu_r})}(X;1,t). \end{align} $$

Finally, we will show that each $\tilde {H}^{1,n}_{(1^{\nu _r})}(X;1,t)$ is essentially a Hall-Littlewood polynomial. Using the particularly simple form of the series $\mathbf {H}_{\nu }$ when $\mu = (d)$ is a single row, Theorem 5.2.1 and (11) yield $q^{-\binom {d}{2}}\omega \hspace {.3mm} \tilde {H}^{1,n}_{(d) }(X;q,t) = H_{(n^d)}(X;q)$ . By Proposition 5.2.5, $\tilde {H}^{1,n}_{\mu }(X;q,t) = \tilde {H}^{1,n}_{\mu ^* }(X;t,q)$ , and thus,

(42) $$ \begin{align} t^{-\binom{d}{2}}\omega \hspace{.3mm} \tilde{H}^{1,n}_{(1^d) }(X;q,t) = H_{(n^d)}(X;t). \end{align} $$

Formula (37) now follows from (41) and (42).

5.3 A raising operator formula for the $m,n$ -Macdonald polynomials

We now give a raising operator formula for $\tilde {H}_\mu ^{m,n}$ which generalizes the raising operator formula for $\tilde {H}_\mu $ in Theorem 3.1.4, and is derived in a similar way. Specifically, we combine the Haglund-Haiman-Loehr formula (Theorem 4.2.2) with an $m,n$ version of Theorem 4.3.1. This latter result requires some notation involving the dilation of a column diagram.

For $\beta \in {\mathbb Z}_{+}^k$ , let $m \beta = (m \beta _1,\dots , m \beta _k)$ . We think of the column diagram $\boldsymbol {m\beta }$ of $m \beta $ as the result of dilating the column diagram $\boldsymbol {\beta }$ of $\beta $ vertically by a factor of m, so that each box of $\boldsymbol {\beta }$ gives rise to m boxes of $\boldsymbol {m\beta }$ . To be more precise, define the map of boxes

(43) $$ \begin{align} \tau \colon \boldsymbol {m\beta} \to \boldsymbol {\beta}, \ \ \ \ (i,j) \mapsto (i,\lfloor (j-1)/m \rfloor + 1). \end{align} $$

Thus, in the dilation process, each box b of $\boldsymbol {\beta }$ gives rise to a set $\tau ^{-1}(b)$ of m boxes of $\boldsymbol {m\beta }$ , called a dilated box, which forms a contiguous subset of a column of $\boldsymbol {m\beta }$ .

As in §3.1, we write ${\boldsymbol {\beta }[1]},\dots , {\boldsymbol {\beta }[d]}$ for the boxes of $\boldsymbol {\beta }$ listed in increasing reading order and ${\boldsymbol {m\beta }[1]},\dots , {\boldsymbol {m\beta }[l]}$ for the boxes of $\boldsymbol {m\beta }$ in increasing reading order, where $d = |\beta |$ and $l = dm$ . Define a map ${\mathsf {s}} \colon V_\beta \to V_{m \beta }$ which takes a vertical domino $({\boldsymbol {\beta }[i]}, {\boldsymbol {\beta }[j]})$ of $\boldsymbol {\beta }$ to the vertical domino $({\boldsymbol {m\beta }[i']}, {\boldsymbol {m\beta }[j']})$ of $\boldsymbol {m\beta }$ such that $\tau ({\boldsymbol {m\beta }[i']}) = {\boldsymbol {\beta }[i]}$ and $\tau ({\boldsymbol {m\beta }[j']}) = {\boldsymbol {\beta }[j]}$ . Note that ${\mathsf {s}}(V_\beta )$ is the set of vertical dominoes of the dilated diagram $\boldsymbol {m\beta }$ which straddle two dilated boxes.

Example 5.3.1. For m = 3 and β = (2, 1, 2), the dilated boxes of $\boldsymbol {m\beta }$ are shown below along with the labeling of the boxes of $\boldsymbol {\beta }$ and $\boldsymbol {m\beta }$ by reading order. To clarify the definitions of $\tau $ and ${\mathsf {s}}$ , note that τ ⁻¹( β [3]) = { mβ [7], mβ [10], mβ [13]} and s(V _β ) = {( mβ [5], mβ [7]), ( mβ [6], mβ [9])}.

Given $(m,n)\in {\mathbb Z} _{+}\times {\mathbb Z}_+ $ , we define the sequence of m integers as in [Reference Blasiak, Haiman, Morse, Pun and Seelinger7, (9.5)]

(44) $$ \begin{align} {\mathbf b} (m,n)_{i} = \lceil i n/m \rceil -\lceil (i-1) n/m \rceil \qquad (i=1,\ldots,m). \end{align} $$

We then define, for any $\beta \in {\mathbb Z}_{+}^k$ of size $d= |\beta |$ , a weight ${\mathbf b}(m,n,\beta ) \in {\mathbb Z}^{dm}$ as follows: fill each dilated box of $\boldsymbol {m\beta }$ with the sequence ${\mathbf b}(m,n)$ from north to south, and then read this filling by the reading order on $\boldsymbol {m\beta }$ . Equivalently, ${\mathbf b}(m,n,\beta )_r = {\mathbf b}(m,n)_a$ , where a is the integer $a\in \{1,\dots ,m\}$ such that $a\equiv -j+1 \pmod {m}$ for j the row index of the r-th box of $\boldsymbol {m\beta }$ in reading order (i.e., ${\boldsymbol {m \beta }[\, r]} = (i,j)$ for some i).

Theorem 5.3.2 [Reference Blasiak, Haiman, Morse, Pun and Seelinger7].

Let $m,n$ be coprime positive integers, let $\beta \in {\mathbb Z}_{+}^k$ , and set $d = |\beta |$ , $l = dm$ . For $S \subseteq V_\beta $ , let ${\boldsymbol \nu }(S)$ be the k-tuple of ribbons in Definition 4.2.1. The action of the LLT polynomial ${\mathcal G}_{{\boldsymbol \nu }(S)}[-MX^{m,n}] \in \Lambda (X^{m,n})$ on 1 is given by

(45) $$ \begin{align} {\mathcal G}_{{\boldsymbol \nu }(S)}[-MX^{m,n}] \cdot 1 = \omega \operatorname{\mathrm{pol}}_X\, & {\boldsymbol \sigma } \bigg( (-qt)^{|V_\beta\setminus S|} q^{A_\beta+(m-1)\mathsf{n}(\beta^+)} \mathbf{z}^{{\mathbf b}(m,n,\beta)} \nonumber\\ &\times \frac{ \prod_{({\boldsymbol {m\beta }[i]}, {\boldsymbol {m\beta }[j]}) \in {\mathsf{s}}(V_\beta \setminus S)} z_i/z_j \prod_{\alpha_{ij} \in \widehat{R}_{m\beta}} \! \big(1-q\hspace{.3mm} t\, z_i/z_j \big)} { \prod_{\alpha_{ij} \in R_{+}(\operatorname{\mathrm{GL}}_{l})} \! \big(1-q \, z_i/z_j\big) \prod_{\alpha_{ij} \in R_{m\beta}} \! \big(1-t \, z_i/z_j\big)}\bigg), \end{align} $$

where $A_\beta $ is the number of attacking pairs of ${\boldsymbol \nu }(S)$ as in §4.3, $\beta ^+$ is the partition rearrangement of $\beta $ , and $R_{m\beta }, \hspace {.3mm} \widehat {R}_{m\beta } \subseteq R_+(\operatorname {\mathrm {GL}}_l)$ are as in (15, 16).

We now give our raising operator formula for the $m,n$ -Macdonald polynomials $\tilde {H}_{\mu }^{m,n}(X;q,t)$ .

Theorem 5.3.3. Let $m,n$ be coprime positive integers. Given $\beta \in {\mathbb Z}_{+}^k$ and setting $d = |\beta |$ and $l = dm$ , define the $m,n$ -Macdonald series by

(46)

which we regard as an infinite formal linear combination of irreducible $\operatorname {\mathrm {GL}} _{l}$ characters using the convention of Remark 2.2.1. Then, for any partition $\mu $ and any rearrangement $\beta $ of $\mu ^{*}$ ,

(47) $$ \begin{align} \tilde{H}_{\mu}^{m,n}(X;q,t) = \omega \operatorname{\mathrm{pol}}_X \! \Big( q^{m \hspace{.3mm} \mathsf{n}(\mu^*)} \hspace{.3mm} t^{\mathsf{n}(\mu)} \hspace{.3mm} \mathbf{H}_{\beta}^{m,n}({\mathbf z};q, t) \Big). \end{align} $$

Remark 5.3.4. (i) In (46), the indices of $z_i$ correspond to the boxes of the dilated diagram $\boldsymbol {m\beta }$ , while the arm and leg are taken with respect to the original diagram $\boldsymbol {\beta }$ .

(ii) Since ${\mathbf b}(1,n,\beta ) = n^l$ , $\mathbf {H}^{1,n}_{\beta } = (z_1\cdots z_l)^n\mathbf {H}_{\beta }$ , where $\mathbf {H}_{\beta }$ is the Macdonald series from Definition 3.1.3. Hence, Theorem 5.3.3 proves Theorem 5.2.1 by setting $m=1$ with n arbitrary. It also specializes to Theorem 3.1.4 when $(m,n)= (1,1)$ .

(iii) Expanding on (ii), the series $\mathbf {H}^{m,n}_{\beta }$ simultaneously encodes the $m,n$ -Macdonald polynomials $\{H^{m,n'}_\mu : n' \in (n+m{\mathbb Z}) \cap {\mathbb Z}_+\}$ , in the following sense:

(48) $$ \begin{align} \tilde{H}_{\mu}^{m,n+am} = \omega \operatorname{\mathrm{pol}}_X \! \Big( q^{m \hspace{.3mm} \mathsf{n}(\mu^*)} \hspace{.3mm} t^{\mathsf{n}(\mu)} \hspace{.3mm} (z_1\cdots z_l)^a \hspace{.3mm} \mathbf{H}_{\beta}^{m,n}\Big). \end{align} $$

This follows from Theorem 5.3.3 using that ${\mathbf b}(m,n+a\hspace {.3mm} m,\beta ) = {\mathbf b}(m,n,\beta ) + a^l$ , which in turn holds by ${\mathbf b}(m,n+a\hspace {.3mm} m) = {\mathbf b}(m,n)+a^m$ .

Theorem 5.3.3 and Remark 5.3.4 (iii) have the following corollary.

Proof of Theorem 5.3.3.

By a plethystic transformation and change of variables we can replace X in (28) with $-MX^{m,n}$ . Letting both sides act on $1 \in \Lambda (X)$ and then substituting in (45) yields

(49) $$ \begin{align} & \tilde{H}_\mu[-M X^{m,n}] \cdot 1 = \omega \operatorname{\mathrm{pol}}_X {\boldsymbol \sigma } \bigg( \frac{\mathbf{z}^{{\mathbf b}(m,n,\beta)} \cdot \Upsilon \cdot \prod_{\alpha_{ij} \in \widehat{R}_{m\beta} } \big(1-q\hspace{.3mm} t\, z_i/z_j \big)} {\prod_{\alpha_{ij} \in R_+} \big(1-q \, z_i/z_j\big) \prod_{\alpha_{ij} \in R_{m\beta}} \big(1-t \, z_i/z_j\big)}\bigg), \end{align} $$

where

(50) $$ \begin{align} \Upsilon = \sum_{S \subseteq V_\beta} (-qt)^{|V_\beta \setminus S|} q^{A_\beta+(m-1)\mathsf{n}(\mu^*)} \prod_{({\boldsymbol {\beta}[i]}, {\boldsymbol {\beta}[j]})\in S} \!\!\! q^{-{\mathrm{arm}}({\boldsymbol {\beta}[i]})} \hspace{.3mm} t^{{\mathrm{leg}}({\boldsymbol {\beta}[i]})+1} \prod_{({\boldsymbol {m\beta}[i]}, {\boldsymbol {m\beta}[j]})\in {\mathsf{s}}(V_\beta \setminus S)} \!\!\!\!z_i/z_j. \end{align} $$

The proof of Theorem 3.1.4 establishes that $A_\beta = \mathsf {n}(\mu ^*)+ \sum _{({\boldsymbol {\beta }[i]},{\boldsymbol {\beta }[j]}) \in V_\beta } {\mathrm {arm}}({\boldsymbol {\beta }[i]})$ , as can also be checked combinatorially. Using this, $\Upsilon $ simplifies essentially the same way it did in that proof:

(51) $$ \begin{align} \Upsilon \ =\ q^{m \hspace{.3mm} \mathsf{n}(\mu^*)} \hspace{.3mm} t^{\mathsf{n}(\mu)} \prod_{({\boldsymbol {m\beta}[i]}, {\boldsymbol {m\beta}[j]}) \in {\mathsf{s}}(V_\beta)} \big(1- q^{{\mathrm{arm}}(\tau({\boldsymbol {m\beta}[i]}))+1}\hspace{.3mm} t^{-{\mathrm{leg}}(\tau({\boldsymbol {m\beta}[i]}))} z_i/z_j \big). \end{align} $$

Plugging this back into (49) completes the proof.

We obtain yet other expressions for the modified Macdonald polynomials from Theorem 5.3.3.

Corollary 5.3.6. Let $\mu $ be a partition of d and $\beta $ a rearrangement of $\mu ^*$ . The modified Macdonald polynomial $\tilde {H}_{\mu }(X;q,t)$ can be expressed in terms of the $m,1$ -Macdonald series for any m as follows:

(52) $$ \begin{align} \tilde{H}_\mu(X;q,t) = \omega \operatorname{\mathrm{pol}}_X \! \big( t^{(-m+1)\mathsf{n}(\mu)}\, \mathbf{H}_{\beta}^{m,1}({\mathbf z};q, t)\big). \end{align} $$