2 The right-action skew Littlewood–Richardson rule for Hopf algebras

We begin by recalling and deducing general Hopf algebra results that will be useful later. Following Tewari and van Willigenburg [Reference Tewari and van Willigenburg23], let H and $H^{*}$ be a pair of dual Hopf algebras over a field k with duality pairing $\langle\hspace{2pt} ,\hspace{2pt} \rangle : H \otimes H^{*} \rightarrow k$ for which the structure of $H^{*}$ is dual to that of H and vice versa. Let $h\in H, a\in H^{*}$ . By Sweedler notation, we have coproduct denoted by $\Delta h=\sum h_1\otimes h_2$ , and similarly $h_1h_2 = h_1\cdot h_2$ denotes product. We define the action of one algebra on the other one by the following:

(2.1) $$ \begin{align} h\rightharpoonup a=\sum\langle h, a_2\rangle a_1, \end{align} $$

(2.2) $$ \begin{align} a\rightharpoonup h=\sum\langle h_2, a\rangle h_1. \end{align} $$

Let $S:H\rightarrow H$ denote the antipode map. Then for $\Delta h= \sum h_1\otimes h_2$ ,

(2.3) $$ \begin{align} \sum(Sh_1)h_2=\varepsilon(h)1_H=\sum h_1(Sh_2), \end{align} $$

where $\varepsilon $ and $1$ denote counit and unit, respectively. Following Montgomery [Reference Montgomery18], we can define the convolution product $*$ for f and g in H by

$$\begin{align*}(f * g)(a)=\sum \langle f,a_1\rangle \langle g,a_2\rangle =\langle fg,a\rangle.\end{align*}$$

Then it follows that

$$\begin{align*}\langle g,f\rightharpoonup a\rangle=\langle gf,a\rangle.\end{align*}$$

Similarly, $\langle a\rightharpoonup f,b\rangle = \langle f,ba\rangle .$ Since $H^{*}$ is a left H-module algebra under $\rightharpoonup $ , we have that

$$\begin{align*}h\rightharpoonup(a\cdot b)=\sum (h_1\rightharpoonup a)\cdot(h_2\rightharpoonup b).\end{align*}$$

Lemma 2.1 [Reference Lam, Lauve and Sottile12]

For $g,h\in H$ and $a\in H^{*}$ ,

$$\begin{align*}(a\rightharpoonup g)\cdot h=\sum(S(h_2)\rightharpoonup a)\rightharpoonup (g\cdot h_1),\end{align*}$$

where $S: H\rightarrow H$ is the antipode.

As in Montgomery [Reference Montgomery18], define a right action by the following:

(2.4) $$ \begin{align} h\leftharpoonup a=\sum \langle h,a_1\rangle a_2, \end{align} $$

(2.5) $$ \begin{align} a\leftharpoonup h = \sum \langle h_1,a\rangle h_2. \end{align} $$

As before, it follows that $\langle g,f\leftharpoonup a\rangle =\langle fg,a\rangle $ and $\langle a\leftharpoonup f,b\rangle =\langle f,ab\rangle $ .

Lemma 2.2 Let $f \in H$ and $a,b \in H^{*}$ . Then

$$\begin{align*}f\leftharpoonup (a\cdot b)= \sum (f_1\leftharpoonup a)\cdot (f_2 \leftharpoonup b).\end{align*}$$

Proof Let $f,g \in H$ and $a,b \in H^{*}$ . Then

$$ \begin{align*} \langle g,f\leftharpoonup (a\cdot b)\rangle &= \langle fg,ab\rangle \\ &=\langle a \leftharpoonup (fg),b\rangle\\ &=\sum\langle f_1g_1,a\rangle \langle f_2g_2,b\rangle\\ &=\sum \langle g_1,f_1\leftharpoonup a\rangle \langle g_2,f_2\leftharpoonup b\rangle \\ &=\sum \langle g, (f_1\leftharpoonup a)\cdot (f_2\leftharpoonup b)\rangle. \end{align*} $$

Thus, $f \leftharpoonup (a\cdot b) = \sum (f_1\leftharpoonup a)\cdot (f_2\leftharpoonup b)$ .

Lemma 2.3 Let $a \in H^{*}$ . Then

$$\begin{align*}\varepsilon(h)\cdot 1_H \leftharpoonup a = a\end{align*}$$

for any $h \in H$ .

Proof Let $a \in H^{*}$ and $h\in H$ . Then

$$\begin{align*}\varepsilon(h)\cdot 1_H \leftharpoonup a = \sum \langle \varepsilon(h)\cdot 1_H,a_1\rangle a_2. \end{align*}$$

This is only nonzero when $a_1=1_{H^{*}}$ .

Lemma 2.4 Let $h \in H$ and $a,b \in H^{*}$ . Then

$$\begin{align*}a\cdot(h\leftharpoonup b) = \sum h_1 \leftharpoonup ( (S(h_2)\leftharpoonup a)\cdot b).\end{align*}$$

Proof Expand the sum using Lemma 2.2 and coassociativity, $(\Delta \otimes 1)\circ \Delta (h) = (1\otimes \Delta )\circ \Delta (h) = \sum h_1\otimes h_2\otimes h_3$ , to get

$$ \begin{align*} \sum h_1 \leftharpoonup ( (S(h_2)\leftharpoonup a)\cdot b)&=\sum (h_1\leftharpoonup (S(h_2)\leftharpoonup a))\cdot (h_3\leftharpoonup b)\\ &=\sum(h_1\cdot S(h_2)\leftharpoonup a) \cdot (h_3\leftharpoonup b) \text{ since } H^{*} \text{ is an } H\text{-module}\\ &=((\varepsilon(h)\cdot 1_H)\leftharpoonup a) \cdot (h\leftharpoonup b) \text{ by ({2.3})}\\ &=a \cdot (h\leftharpoonup b) \text{ by Lemma~{2.3}}. \\[-36pt] \end{align*} $$

Lemma 2.5 Let $g,h \in H$ and $a \in H^{*}$ . Then

$$\begin{align*}h\cdot (a\leftharpoonup g) = \sum (S(h_2) \leftharpoonup a) \leftharpoonup (h_1\cdot g).\end{align*}$$

Proof Let $g,h \in H$ and $a,b \in H^{*}$ . Then

$$ \begin{align*} \langle h\cdot (a\leftharpoonup g),b\rangle &= \langle a\leftharpoonup g, h\leftharpoonup b\rangle \\ &= \langle g,a\cdot (h\leftharpoonup b)\rangle\\ &=\left \langle g, \sum(h_1\leftharpoonup ((S(h_2)\leftharpoonup a)\cdot b)\right\rangle \text{ by Lemma {2.4}}\\ &=\sum \langle g,h_1\leftharpoonup ((S(h_2)\leftharpoonup a)\cdot b)\rangle\\ &=\sum\langle h_1\cdot g,(S(h_2)\leftharpoonup a)\cdot b \rangle\\ &=\sum \langle (S(h_2)\leftharpoonup a)\leftharpoonup (h_1\cdot g), b\rangle. \\[-35pt] \end{align*} $$

We can use the right action to obtain an algebraic Littlewood–Richardson formula analogous to [Reference Lam, Lauve and Sottile12, Theorem 2.1] for those bases whose skew elements appear as the right tensor factor in the coproduct.

Let $\{L_\alpha \} \subset H$ and $\{R_\beta \} \subset H^{*}$ be dual bases with indexing set $\mathcal {P}$ . Then

(2.6) $$ \begin{align} L_\alpha \cdot L_\beta = \sum_{\gamma}b^\gamma_{\alpha,\beta}L_\gamma & \qquad \Delta(L_\gamma) = \sum_{\alpha,\beta} c^\gamma_{\alpha,\beta} L_\alpha \otimes L_\beta, \end{align} $$

(2.7) $$ \begin{align} R_\alpha\cdot R_\beta = \sum_{\gamma}c^\gamma_{\alpha,\beta}R_\gamma & \qquad \Delta(R_\gamma) = \sum_{\alpha,\beta}b^\gamma_{\alpha, \beta} R_\alpha \otimes R_\beta, \end{align} $$

where $b^\gamma _{\alpha ,\beta }$ and $c^\gamma _{\alpha ,\beta }$ are structure constants. We can also write

(2.8) $$ \begin{align} \Delta(L_\gamma) = \sum_{\delta} L_\delta \otimes L_{\gamma/\delta} \qquad \Delta(R_\gamma) = \sum_{\delta} R_\delta \otimes R_{\gamma/\delta}. \end{align} $$

Note that $L_\alpha \leftharpoonup R_\beta = {R_{\beta /\alpha }}$ and $R_\beta \leftharpoonup L_\alpha = L_{\alpha /\beta }$ . Further,

(2.9) $$ \begin{align} \Delta(L_{\alpha/\beta}) = \sum_{\pi,\rho}c^\alpha_{\pi,\rho,\beta} L_\pi \otimes L_\rho \qquad \Delta(R_{\alpha/\beta}) = \sum_{\pi,\rho}b^\alpha_{\pi,\rho,\beta}R_\pi \otimes R_\rho. \end{align} $$

The antipode acts on $L_\rho $ by $S(L_\rho ) = (-1)^{\theta (\rho )}L_{\rho ^{*}}$ where $\theta :\mathcal {P}\rightarrow \mathbb {N}$ and $*:\mathcal {P}\rightarrow \mathcal {P}$ .

Theorem 2.6 For $\alpha , \beta ,\gamma ,\delta \in \mathcal {P}$ ,

$$\begin{align*}L_{\alpha/\beta}\cdot L_{\gamma/\delta} = \sum_{\pi,\rho,\nu,\mu}(-1)^{\theta(\rho)}c^\alpha_{\pi,\rho,\beta}b^\nu_{\pi,\gamma}b^\delta_{\mu,\rho^{*}} L_{\nu/\mu}.\end{align*}$$

Proof We use Lemma 2.5 and the preceding facts about the product, coproduct, and antipode maps on H and $H^{*}$ to obtain

$$ \begin{align*} L_{\alpha/\beta}\cdot L_{\gamma/\delta} & = L_{\alpha/\beta}\cdot (R_\delta \leftharpoonup L_\gamma)\\ &=\sum_{\pi,\rho} c^\alpha_{\pi,\rho,\beta}(S(L_\rho)\leftharpoonup R_\delta) \leftharpoonup (L_\pi\cdot L_\gamma)\\ &= \sum_{\pi,\rho}(-1)^{\theta(\rho)}c^\alpha_{\pi,\rho,\beta}(L_{\rho^{*}} \leftharpoonup R_\delta) \leftharpoonup(L_\pi\cdot L_\gamma)\\ &=\sum_{\pi,\rho}(-1)^{\theta(\rho)}c^\alpha_{\pi,\rho,\beta}\left(R_{\delta/\rho^{*}} \leftharpoonup \left(\sum_{\nu}b^\nu_{\pi,\gamma}L_\nu\right)\right)\\ &=\sum_{\pi,\rho,\nu}(-1)^{\theta(\rho)}c^\alpha_{\pi,\rho,\beta}b^\nu_{\pi,\gamma} (R_{\delta/\rho^{*}}\leftharpoonup L_\nu)\\ &=\sum_{\pi,\rho,\nu,\mu}(-1)^{\theta(\rho)}c^\alpha_{\pi,\rho,\beta}b^\nu_{\pi,\gamma} b^\delta_{\mu,\rho^{*}}(R_\mu\leftharpoonup L_\nu)\\ &=\sum_{\pi,\rho,\nu,\mu}(-1)^{\theta(\rho)}c^\alpha_{\pi,\rho,\beta}b^\nu_{\pi,\gamma} b^\delta_{\mu,\rho^{*}} L_{\nu/\mu}. \\[-42pt] \end{align*} $$

3 The dual Hopf algebras $\operatorname {QSym}$ and $\operatorname {NSym}$

We now focus our attention on the dual Hopf algebra pair of noncommutative symmetric functions and quasisymmetric functions, and introduce our main objects of study the (row-strict) dual immaculate functions.

A composition $\alpha = (\alpha _1, \ldots , \alpha _k)$ of n, denoted by $\alpha \vDash n$ is a list of positive integers such that $\sum _ {i=1} ^{k} \alpha _i = n$ . We call n the size of $\alpha $ and sometimes denote it by $|\alpha |$ , and call k the length of $\alpha $ and sometimes denote it by $\ell (\alpha )$ . If $\alpha _{j_1}= \cdots = \alpha _{j_m} = i$ , we sometimes abbreviate this to $i^m$ , and denote the empty composition of 0 by $\emptyset $ . There exists a natural correspondence between compositions $\alpha \vDash n$ and subsets $S\subseteq \{ 1, \ldots , n-1\} = [n-1]$ . More precisely, $\alpha = (\alpha _1, \ldots , \alpha _k)$ corresponds to $\operatorname {\mathrm {set}} (\alpha ) = \{ \alpha _1, \alpha _1 + \alpha _2, \ldots , \alpha _1 + \cdots +\alpha _{k-1}\}$ , and conversely $S= \{ s_1, \ldots , s_{k-1}\}$ corresponds to $\operatorname {\mathrm {comp}} (S) = ( s_1, s_2 - s_1, \ldots , n- s_{k-1})$ . We also denote by $S^c$ the set complement of S in $[n-1]$ .

Given a composition $\alpha $ , its diagram, also denoted by $\alpha $ , is the array of left-justified boxes with $\alpha _i$ boxes in row i from the bottom. Given two compositions $\alpha , \beta $ we say that $\beta \subseteq \alpha $ if $\beta _j \leq \alpha _j$ for all $1\leq j \leq \ell (\beta ) \leq \ell (\alpha )$ , and given $\alpha , \beta $ such that $\beta \subseteq \alpha $ , the skew diagram $\alpha / \beta $ is the array of boxes in $\alpha $ but not $\beta $ when $\beta $ is placed in the bottom-left corner of $\alpha $ . If, furthermore, $\beta \subseteq \alpha $ and $\alpha _j - \beta _j \in \{ 0,1\}$ for all $1\leq j \leq \ell (\beta )\leq \ell (\alpha ), $ then we call $\alpha / \beta $ a vertical strip.

Example 3.1 If $\alpha = (3,4,1)$ , then $|\alpha |=8, \ell (\alpha ) = 3$ , and $\operatorname {\mathrm {set}} (\alpha ) = \{3,7\}$ . Its diagram is

and if $\beta = (2,4)$ , then

is a vertical strip.

Given a standard (skew) immaculate tableau, T, its descent set is

$$ \begin{align*} \operatorname{\mathrm{Des}} (T) = \{ i : i+1 \mbox{ appears strictly above } i \mbox{ in } T \}. \end{align*} $$

Example 3.3 A standard skew immaculate tableau of shape $(3,4,1)/(1)$ is

with $\operatorname {\mathrm {Des}} (T) = \{ 1,5,6 \}$ .

We are now ready to define our Hopf algebras and functions of central interest.

Given a composition $\alpha = ( \alpha _1, \ldots , \alpha _k) \vDash n$ and commuting variables $\{ x_1, x_2, \ldots \}$ we define the monomial quasisymmetric function $M_\alpha $ to be

$$ \begin{align*}M_\alpha = \sum _{i_1 < \cdots < i_k} x_{i_1} ^{\alpha _1} \cdots x_{i_k} ^{\alpha _k}\end{align*} $$

the fundamental quasisymmetric function $F_\alpha $ to be

$$ \begin{align*}F_\alpha = \sum _{i_1 \leq \cdots \leq i_n \atop i_j = i_{j+1} \Rightarrow j \not\in \operatorname{\mathrm{set}}(\alpha)} x_{i_1} \cdots x_{i_n}\end{align*} $$

the dual immaculate function $\mathfrak {S}^{*}_\alpha $ to be

$$ \begin{align*}\mathfrak{S}^{*} _\alpha = \sum _{T} F_{\operatorname{\mathrm{comp}}(\operatorname{\mathrm{Des}}(T))}\end{align*} $$

and the row-strict dual immaculate function $\mathcal {R}\mathfrak {S}^{*} _\alpha $ to be

$$ \begin{align*}\mathcal{R}\mathfrak{S}^{*} _\alpha = \sum _{T} F_{\operatorname{\mathrm{comp}}(\operatorname{\mathrm{Des}}(T)^c),}\end{align*} $$

where the latter two sums are over all standard immaculate tableaux T of shape $\alpha $ . These extend naturally to give skew dual immaculate and row-strict dual immaculate functions $\mathfrak {S}^{*} _{\alpha / \beta }$ [Reference Berg, Bergeron, Saliola, Serrano and Zabrocki6] and $\mathcal {R}\mathfrak {S}^{*} _{\alpha / \beta }$ [Reference Niese, Sundaram, van Willigenburg, Vega and Wang19], where $\alpha / \beta $ is a skew diagram.

Example 3.4 We have that $M_{(2)} = x^2 _1 + x_2 ^2 + x_3 ^2 + \cdots $ and $F_{(2)} = x^2 _1 + x_2 ^2 + x_3 ^2 + \cdots + x_1x_2+x_1x_3+x_2x_3+ \cdots = \mathfrak {S}^{*} _{(2)} = \mathcal {R}\mathfrak {S}^{*} _{(1^2)}$ from the following standard immaculate tableau T with $\operatorname {\mathrm {Des}} (T) = \emptyset $ .

The set of all monomial or fundamental quasisymmetric functions forms a basis for the Hopf algebra of quasisymmetric functions $\operatorname {\mathrm {QSym}}$ , as does the set of all (row-strict) dual immaculate functions. There exists an involutory automorphism $\psi $ defined on fundamental quasisymmetric functions by

$$ \begin{align*}\psi (F_\alpha) = F_{\operatorname{\mathrm{comp}} (\operatorname{\mathrm{set}} (\alpha ^c))}\end{align*} $$

such that [Reference Niese, Sundaram, van Willigenburg, Vega and Wang19]

$$ \begin{align*} \psi (\mathfrak{S}^{*}_\alpha) = \mathcal{R}\mathfrak{S}^{*} _\alpha \end{align*} $$

for a composition $\alpha $ . This extends naturally to skew diagrams $\alpha / \beta $ to give

$$ \begin{align*} \psi (\mathfrak{S}^{*} _{\alpha/\beta}) = \mathcal{R}\mathfrak{S}^{*} _{\alpha/\beta}. \end{align*} $$

Dual to the Hopf algebra of quasisymmetric functions is the Hopf algebra of noncommutative symmetric functions $\operatorname {NSym}$ . Given a composition $\alpha = (\alpha _1, \ldots , \alpha _k) \vDash n$ and noncommuting variables $\{ y_1, y_2, \ldots \}$ we define the nth elementary noncommutative symmetric function $\boldsymbol {e}_n$ to be

$$ \begin{align*} \boldsymbol{e}_n = \sum_{i_1<\cdots < i_n} y_{i_1}\cdots y_{i_n} \end{align*} $$

and the elementary noncommutative symmetric function $\boldsymbol {e}_\alpha $ to be

$$ \begin{align*} \boldsymbol{e}_\alpha = \boldsymbol{e}_{\alpha _1} \cdots \boldsymbol{e}_{\alpha _k}. \end{align*} $$

Meanwhile, we define the nth complete homogeneous noncommutative symmetric function $\boldsymbol {h} _n$ to be

$$ \begin{align*}\boldsymbol{h} _n = \sum _{i_1\leq \cdots \leq i_n}y_{i_1}\cdots y_{i_n} \end{align*} $$

and the complete homogeneous noncommutative symmetric function $\boldsymbol {h} _\alpha $ to be

$$ \begin{align*}\boldsymbol{h}_\alpha = \boldsymbol{h} _{\alpha _1} \cdots \boldsymbol{h} _{\alpha _k}.\end{align*} $$

The set of all elementary or complete homogeneous noncommutative symmetric functions forms a basis for $\operatorname {NSym}$ . The duality between $\operatorname {QSym}$ and $\operatorname {NSym}$ is given by

$$ \begin{align*}\langle M_\alpha , \boldsymbol{h} _\alpha \rangle = \delta _{\alpha\beta,}\end{align*} $$

where $\delta _{\alpha \beta }=1$ if $\alpha = \beta $ and 0 otherwise. This induces the bases dual to the (row-strict) dual immaculate functions via

$$ \begin{align*}\langle \mathfrak{S}^{*} _\alpha , {\mathfrak{S}} _\alpha \rangle = \delta _{\alpha\beta} \qquad \langle \mathcal{R}\mathfrak{S}^{*}_\alpha , \mathcal{R}{\mathfrak{S}} _\alpha \rangle = \delta _{\alpha\beta}\end{align*} $$

and implicitly defines the bases of immaculate and row-strict immaculate functions. While concrete combinatorial definitions of these functions have been established [Reference Berg, Bergeron, Saliola, Serrano and Zabrocki6, Reference Niese, Sundaram, van Willigenburg, Vega and Wang19], we will not need them here. However, what we will need is the involutory automorphism in $\operatorname {NSym}$ corresponding to $\psi $ in $\operatorname {QSym}$ , defined by $\psi (\boldsymbol {e}_\alpha ) = \boldsymbol {h} _\alpha $ that gives $\psi ({\mathfrak {S}} _\alpha ) = \mathcal {R}{\mathfrak {S}} _\alpha $ [Reference Niese, Sundaram, van Willigenburg, Vega and Wang19].

4 The Pieri rules for skew dual immaculate functions

A left Pieri rule for immaculate functions was conjectured in [Reference Berg, Bergeron, Saliola, Serrano and Zabrocki6, Conjecture 3.7] and proved in [Reference Bergeron, Sánchez-Ortega and Zabrocki8]. Given a composition $\alpha = (\alpha _1,\ldots ,\alpha _k),$ we say that $\operatorname {\mathrm {tail}}(\alpha ) = (\alpha _2,\ldots ,\alpha _k)$ . If $\beta \in \mathbb {Z}^k$ , then $\operatorname {\mathrm {neg}}(\alpha -\beta ) = |\{i:\alpha _i-\beta _i<0\}|$ . Let $\operatorname {\mathrm {sgn}}(\beta )=(-1)^{\operatorname {\mathrm {neg}}(\beta )}$ with $\operatorname {\mathrm {neg}}(\beta )=|\{i:\beta _i<0\}|$ .

Following [Reference Bergeron, Sánchez-Ortega and Zabrocki8], we define $Z_{s,\alpha }$ to be a set of all $\beta \in \mathbb {Z}^k$ such that:

1. (1) $\beta _1+\cdots +\beta _k=s$ and $\beta _1+\cdots +\beta _i\leq s$ for all $i<k$ .

2. (2) $\alpha _i-\beta _i\geq 0$ for all $1\leq i\leq k$ and $|i:\alpha _i-\beta _i=0|\leq 1$ .

3. (3) For all $1\leq i\leq k$ ,

   • • if $\alpha _i>s-(\beta _1+\cdots +\beta _{i-1}),$ then $ 0\leq \beta _i\leq s-(\beta _1+\cdots +\beta _{i-1}),$

   • • if $\alpha _i<s-(\beta _1+\cdots +\beta _{i-1})$ , then $\beta _i<0$ , and

   • • if $\alpha _i=s-(\beta _1+\cdots +\beta _{i-1}),$ then either $\beta _i<0$ or $\beta _i=\alpha _i$ and $\beta _{i+1}=\cdots = \beta _k=0$ .

Now we are ready to define the coefficients of the immaculate basis appearing in the left Pieri rule.

Definition 4.1 [Reference Bergeron, Sánchez-Ortega and Zabrocki8]

For a positive integer s and compositions $\alpha , \gamma $ with $|\alpha |-|\gamma |=s$ , let $1\leq j\leq k$ be the smallest integer such that $\alpha _i=\gamma _{i-1}$ for all $j<i\leq k$ where $j=k$ when $\alpha _k\neq \gamma _{k-1}$ . Let $j\leq r\leq k$ be the largest integer such that $\alpha _j<\alpha _{j+1}<\cdots <\alpha _r$ . Let $\alpha ^{(i)} = (\alpha _1,\ldots ,\alpha _i).$ Then define

$$\begin{align*}c^{\gamma}_{s,\alpha}= \left\{\begin{array}{@{}ll}\operatorname{\mathrm{sgn}}(\alpha-\gamma),&\text{if } \ell(\gamma)=\ell(\alpha) \text{ and } \alpha-\gamma\in Z_{s,\alpha},\\ \operatorname{\mathrm{sgn}}(\alpha^{(j-1)}-\gamma^{(j-1)}), & \text{if } \ell(\gamma)=\ell(\alpha)-1,\\ &r-j \text{ is even, and } \\ &(\alpha^{(j-1)}-\gamma^{(j-1)},\alpha_j,0,\ldots,0)\in Z_{s,\alpha}, \\ 0,&\text{otherwise.}\end{array} \right. \end{align*}$$

Theorem 4.2 [Reference Berg, Bergeron, Saliola, Serrano and Zabrocki6, Reference Bergeron, Sánchez-Ortega and Zabrocki8]

Let $m>0$ and $\alpha $ be a composition. Then

$$\begin{align*}\boldsymbol{h}_m{\mathfrak{S}}_\alpha = \sum_{\substack{\beta\vDash |\alpha|+m\\\beta_1\geq m\\0\leq \ell(\beta)-\ell(\alpha)\leq 1}} c^{\operatorname{\mathrm{tail}}(\beta)}_{\beta_1-m, \alpha} {\mathfrak{S}}_\beta. \end{align*}$$

Applying $\psi $ to both sides of the left Pieri rule in Theorem 4.2 immediately yields a left Pieri rule for row-strict immaculate functions.

Corollary 4.3 Let $m>0$ and $\alpha $ be a composition. Then

$$\begin{align*}\boldsymbol{e}_m \mathcal{R}{\mathfrak{S}}_\alpha = \sum_{\substack{\beta\vDash |\alpha|+m\\ \beta_1\geq m\\ 0\leq \ell(\beta)-\ell(\alpha)\leq 1}}c^{\operatorname{\mathrm{tail}}(\beta)}_{\beta_1-m, \alpha} \mathcal{R}{\mathfrak{S}}_\beta. \end{align*}$$

Lemma 3.1 of [Reference Bergeron, Sánchez-Ortega and Zabrocki8] shows that for $s\geq 0$ , $r>0$ and compositions $\alpha ,\beta $ with $|\alpha |=|\beta |+s$ ,

$$\begin{align*}\langle {\mathfrak{S}}_\alpha,{F_{(s)}}\mathfrak{S}^{*}_\beta\rangle=\langle \boldsymbol{h}_r{\mathfrak{S}}_\alpha,\mathfrak{S}^{*}_{(s+r,\beta)}\rangle.\end{align*}$$

This leads to the following Pieri rule for dual immaculate functions.

Theorem 4.4 [Reference Bergeron, Sánchez-Ortega and Zabrocki8]

Let $s>0$ and $\alpha $ be a composition. Then

$$\begin{align*}{F_{(s)}}\mathfrak{S}^{*}_\alpha = \sum_{\substack{\beta\vDash |\alpha|+s\\0\leq \ell(\beta)-\ell(\alpha)\leq 1}} c^{\alpha}_{s,\beta} \mathfrak{S}^{*}_\beta.\end{align*}$$

Again, applying $\psi $ to both sides gives a Pieri rule for row-strict dual immaculate functions.

Corollary 4.5 Let $s>0$ and $\alpha $ be a composition. Then

$$\begin{align*}F_{(1^s)} \mathcal{R}\mathfrak{S}^{*}_\alpha = \sum_{\substack{\beta\vDash |\alpha|+s\\0\leq \ell(\beta)-\ell(\alpha)\leq 1}} c^{\alpha}_{s,\beta} \mathcal{R}\mathfrak{S}^{*}_\beta.\\[-24pt]\end{align*}$$

We use these results together with Hopf algebra computations to construct a Pieri rule for skew dual immaculate functions. Using the map $\psi $ , this also gives a Pieri rule for row-strict skew dual immaculate functions. But first, we have a small, yet crucial, lemma.

Proof Recall that if $H= \operatorname {\mathrm {QSym}}$ and $H^{*}=\operatorname {\mathrm {NSym}}$ are our pair of dual Hopf algebras, then we know $\Delta \mathfrak {S}^{*}_\alpha =\sum _\beta \mathfrak {S}^{*}_\beta \otimes \mathfrak {S}^{*}_{\alpha /\beta }$ and we have that

$$\begin{align*}{\mathfrak{S}}_\gamma \leftharpoonup \mathfrak{S}^{*}_\alpha = \sum_\beta \langle {\mathfrak{S}}_\gamma , \mathfrak{S}^{*}_\beta \rangle \mathfrak{S}^{*}_{\alpha/\beta} = \mathfrak{S}^{*}_{\alpha/\gamma}\end{align*}$$

since $\langle {\mathfrak {S}}_\gamma , \mathfrak {S}^{*}_\beta \rangle = \delta _{\gamma \beta }$ , where $\delta _{\gamma \beta } = 1$ if $\gamma =\beta $ and 0 otherwise.

We can now give our Pieri rule for (row-strict) skew dual immaculate functions.

Theorem 4.7 Let $\gamma \subseteq \alpha $ . Then

$$\begin{align*}\mathfrak{S}^{*}_{(s)}\mathfrak{S}^{*}_{\alpha/\gamma}=\sum_{\beta/\tau} (-1)^{|\gamma|-|\tau|}\cdot c^{\alpha}_{|\beta|-|\alpha|,\beta}\,\mathfrak{S}^{*}_{\beta/\tau,}\end{align*}$$

and hence by applying $\psi $ to both sides

$$\begin{align*}\mathcal{R}\mathfrak{S}^{*}_{(s)}\mathcal{R}\mathfrak{S}^{*}_{\alpha/\gamma}=\sum_{\beta/\tau} (-1)^{|\gamma|-|\tau|}\cdot c^{\alpha}_{|\beta|-|\alpha|,\beta}\,\mathcal{R}\mathfrak{S}^{*}_{\beta/\tau,}\end{align*}$$

where $|\beta /\tau |=|\alpha /\gamma |+s$ , $\gamma /\tau $ is a vertical strip of length at most s, $\ell (\beta )-\ell (\alpha ) \in \{0,1\}$ and $c^{\alpha }_{|\beta |-|\alpha |,\beta }$ is the coefficient of Definition 4.1. These decompositions are multiplicity-free up to sign.

Proof Note that $\mathfrak {S}^{*}_{(1^s)}=F_{(1^s)}$ and $\mathfrak {S}^{*}_{(s)}=F_{(s)}$ . Recall that

(4.1) $$ \begin{align} \Delta F_\alpha = \sum_{\substack{(\beta,\gamma) \text{ with }\\ \beta\cdot \gamma = \alpha \text{ or}\\ \beta\odot \gamma=\alpha}} F_\beta\otimes F_\gamma, \end{align} $$

where for $\beta = (\beta _1,\ldots , \beta _k)$ and $\gamma = (\gamma _1,\ldots , \gamma _l)$ , $\beta \cdot \gamma = (\beta _1,\ldots , \beta _k,\gamma _1,\ldots , \gamma _l)$ is the concatenation of $\beta $ and $\gamma $ , and $\beta \odot \gamma = (\beta _1, \ldots , \beta _{k-1},\beta _k+\gamma _1,\gamma _2,\ldots , \gamma _l)$ is the near-concatenation of $\beta $ and $\gamma $ .

Then we have that

$$\begin{align*}\Delta(F_{(s)})=\sum_{i=0}^sF_{(i)}\otimes {F_{(s-i)}.} \end{align*}$$

Thus,

$$ \begin{align*} \mathfrak{S}^{*}_{(s)}\mathfrak{S}^{*}_{\alpha/\gamma} &=\mathfrak{S}^{*}_{(s)}({\mathfrak{S}}_\gamma\leftharpoonup\mathfrak{S}^{*}_\alpha)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{by Lemma 4.6}\\ &=F_{(s)}({\mathfrak{S}}_\gamma\leftharpoonup\mathfrak{S}^{*}_\alpha) \\ &=\sum_{i=0}^s(S({F_{(s-i)}})\leftharpoonup {\mathfrak{S}}_\gamma)\leftharpoonup({F_{(i)}}\mathfrak{S}^{*}_\alpha)\,\,\,\,\,\,\, \text{by Lemma 2.5.} \\ \end{align*} $$

We first compute ${S(F_{(s-i)})}\leftharpoonup {\mathfrak {S}}_\gamma $ . Since it is well known that $S(F_\alpha )=(-1)^{|\alpha |} F_{\operatorname {\mathrm {comp}}(\operatorname {\mathrm {set}}(\alpha )^c)}$ we have that ${S(F_{(s-i)})=(-1)^{s-i}F_{(1^{s-i})}}$ . Furthermore, we can write the coproduct as

$$\begin{align*}\Delta({\mathfrak{S}}_\gamma)=\sum_{\delta,\tau}b^\gamma_{\delta,\tau}{\mathfrak{S}}_\delta\otimes{\mathfrak{S}}_\tau.\end{align*}$$

Thus,

$$ \begin{align*} S(F_{(s-i)})\leftharpoonup {\mathfrak{S}}_\gamma &=(-1)^{s-i}F_{(1^{s-i})}\leftharpoonup {\mathfrak{S}}_\gamma \\ &=\sum_{\delta,\tau}(-1)^{s-i}b^\gamma_{\delta,\tau}\langle F_{(1^{s-i})}, {\mathfrak{S}}_\delta\rangle{\mathfrak{S}}_\tau \\ &=\sum_{\delta,\tau}(-1)^{s-i}b^\gamma_{\delta,\tau}\langle{\mathfrak{S}}^{*}_{(1^{s-i})}, {\mathfrak{S}}_\delta\rangle{\mathfrak{S}}_\tau \\ &=\sum_{\tau}(-1)^{s-i}b^\gamma_{(1^{s-i}), \tau}{\mathfrak{S}}_\tau. \end{align*} $$

By the definition of product and coproduct on $\operatorname {NSym}$ , we have that

$$\begin{align*}b^\gamma_{\delta,\tau}=\langle \Delta{\mathfrak{S}}_\gamma,\mathfrak{S}^{*}_\delta\otimes\mathfrak{S}^{*}_\tau\rangle=\langle{\mathfrak{S}}_\gamma,\mathfrak{S}^{*}_\delta\cdot\mathfrak{S}^{*}_\tau\rangle{.}\end{align*}$$

To compute this for ${\delta = (1^{s-i}),}$ we use Proposition 3.34 from [Reference Berg, Bergeron, Saliola, Serrano and Zabrocki6] which states that $F^\perp _{(1^r)}{\mathfrak {S}}_\alpha = \sum _\beta {\mathfrak {S}}_\beta $ , where $\beta \in \mathbb {Z}^{\ell (\alpha )}$ , $\alpha _k - \beta _k \in \{0,1\}$ for all k and $|\beta | = |\alpha |-r$ . The operator $F^\perp $ is used throughout [Reference Berg, Bergeron, Saliola, Serrano and Zabrocki6], and has the property that $\langle F^\perp {\mathfrak {S}}_\alpha ,\mathfrak {S}^{*}_\beta \rangle = \langle {\mathfrak {S}}_\alpha ,F \mathfrak {S}^{*}_\beta \rangle $ .

Thus,

$$ \begin{align*} b^\gamma_{(1^{s-i}), \tau} &=\langle {\mathfrak{S}}_\gamma, \mathfrak{S}^{*}_{(1^{s-i})}\mathfrak{S}^{*}_\tau\rangle\\ &=\langle{\mathfrak{S}}_\gamma,F_{(1^{s-i})}\mathfrak{S}^{*}_\tau\rangle\\ &=\langle F^{\perp}_{(1^{s-i})}{\mathfrak{S}}_\gamma, \mathfrak{S}^{*}_\tau\rangle\\ &=\left\langle \sum_{\beta}{\mathfrak{S}}_\beta, \mathfrak{S}^{*}_\tau\right\rangle\\ &=\delta _{\beta \tau}, \end{align*} $$

where the sum is over all $\beta $ such that $\beta \in \mathbb {Z}^{\ell (\gamma )}$ , $\gamma _k - \beta _k \in \{0,1\}$ for all k, and $|\beta |=|\gamma |-{(s-i)}$ .

Then using the above calculations, Theorem 4.4 and Lemma 4.6, we have that

$$ \begin{align*} \mathfrak{S}^{*}_{(s)}\mathfrak{S}^{*}_{\alpha/\gamma} &=\mathfrak{S}^{*}_{(s)}({\mathfrak{S}}_\gamma\leftharpoonup\mathfrak{S}^{*}_\alpha)\\ &=\sum_{i=0}^s\left((S(F_{(s-i)})\leftharpoonup {\mathfrak{S}}_\gamma)\leftharpoonup(F_{(i)}\mathfrak{S}^{*}_\alpha)\right)\\ &=\sum_{i=0}^s \left((-1)^{(s-i)}\sum_{\substack{\tau \in \mathbb{Z}^{\ell(\gamma)}\\\gamma_k-\tau_k \in \{0,1\}\\|\tau|=|\gamma|-(s-i)}} {\mathfrak{S}}_\tau\right)\leftharpoonup \left(\sum_{\substack{\beta\vDash |\alpha|+i\\0\leq\ell(\beta) -\ell(\alpha)\leq 1}} c^{\alpha}_{i,\beta} \mathfrak{S}^{*}_{\beta}\right)\\ &=\sum_{i=0}^s \sum_{\substack{\tau,\beta\\ \tau \in \mathbb{Z}^{\ell(\gamma)}\\\gamma_k-\tau_k\in\{0,1\}\\|\tau|=|\gamma|-(s-i)\\\beta\vDash|\alpha|+i\\ \ell(\beta) - \ell(\alpha) \in \{0,1\}}} (-1)^{(s-i)}\cdot c^{\alpha}_{i,\beta}\, \mathfrak{S}^{*}_{\beta/\tau}\\ &=\sum_{\beta/\tau} (-1)^{|\gamma|-|\tau|}\cdot c^{\alpha}_{|\beta|-|\alpha|,\beta}\,\mathfrak{S}^{*}_{\beta/\tau,} \end{align*} $$

where $|\beta /\tau |=|\alpha /\gamma |+s$ , $\gamma /\tau $ is a vertical strip of length at most s, and $\ell (\beta )-\ell (\alpha ) \in \{0,1\}$ .

Example 4.8 Let us compute $\mathfrak {S}^{*}_{(2)}\cdot \mathfrak {S}^{*}_{(1,2,1)/(1,1)}$ .

First, we need to compute all compositions $\beta \vDash 4+i$ for $i\in \{0,1,2\}$ and $\ell (\beta )=3$ or $4$ . We list all possible choices for $\beta $ as the set

$$ \begin{align*} A=\{&(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(1,1,1,2),(1,1,2,1),(1,2,1,1),\\ &(2,1,1,1),(1,1,3),(1,2,2),(1,3,1),(2,1,2),(2,2,1),(3,1,1),(1,1,1,3),\\ &(1,1,2,2),(1,1,3,1),(1,2,1,2),(1,2,2,1),(1,3,1,1),(2,1,1,2),\\ &(2,1,2,1),(2,2,1,1),(3,1,1,1),(1,1,4),(1,2,3),(1,3,2),(1,4,1),\\ &(2,1,3),(2,2,2),(2,3,1),(3,1,2),(3,2,1),(4,1,1)\}. \end{align*} $$

Next, we need to find $\tau $ by removing a vertical strip of length at most $s=2$ from $\gamma =(1,1)$ . We list all options for $\tau $ as the set $B=\{\emptyset ,(1),(1,1)\}$ .

By Theorem 4.7, now we expand $\mathfrak {S}^{*}_{(2)}\cdot \mathfrak {S}^{*}_{(1,2,1)/(1,1)}$ by finding all valid pairs $(\beta ,\tau )$ such that $|\beta /\tau |=4$ . Thus,

$$ \begin{align*} \mathfrak{S}^{*}_{(2)}\cdot\mathfrak{S}^{*}_{(1,2,1)/(1,1)} &=c^{(1,2,1)}_{0,(1,1,1,1)}\mathfrak{S}^{*}_{(1,1,1,1)}+c^{(1,2,1)}_{0,(1,1,2)}\mathfrak{S}^{*}_{(1,1,2)}\\ &\quad +c^{(1,2,1)}_{0,(1,2,1)}\mathfrak{S}^{*}_{(1,2,1)} +c^{(1,2,1)}_{0,(2,1,1)}\mathfrak{S}^{*}_{(2,1,1)}\\ &\quad -c^{(1,2,1)}_{1,(1,1,1,2)}\mathfrak{S}^{*}_{(1,1,1,2)/(1)}- c^{(1,2,1)}_{1,(1,1,2,1)}\mathfrak{S}^{*}_{(1,1,2,1)/(1)}\\ &\quad -c^{(1,2,1)}_{1,(1,2,1,1)}\mathfrak{S}^{*}_{(1,2,1,1)/(1)}- c^{(1,2,1)}_{1,(2,1,1,1)}\mathfrak{S}^{*}_{(2,1,1,1)/(1)}\\ &\quad -c^{(1,2,1)}_{1,(1,1,3)}\mathfrak{S}^{*}_{(1,1,3)/(1)} -c^{(1,2,1)}_{1,(1,2,2)}\mathfrak{S}^{*}_{(1,2,2)/(1)}\\ &-c^{(1,2,1)}_{1,(1,3,1)}\mathfrak{S}^{*}_{(1,3,1)/(1)}-c^{(1,2,1)}_{1,(2,1,2)}\mathfrak{S}^{*}_{(2,1,2)/(1)}\\ &-c^{(1,2,1)}_{1,(2,2,1)}\mathfrak{S}^{*}_{(2,2,1)/(1)}- c^{(1,2,1)}_{1,(3,1,1)}\mathfrak{S}^{*}_{(3,1,1)/(1)}\\ &+c^{(1,2,1)}_{2,(1,1,1,3)}\mathfrak{S}^{*}_{(1,1,1,3)/(1,1)}+c^{(1,2,1)}_{2,(1,1,2,2)} \mathfrak{S}^{*}_{(1,1,2,2)/(1,1)}\\&+c^{(1,2,1)}_{2,(1,1,3,1)}\mathfrak{S}^{*}_{(1,1,3,1)/(1,1)}+ c^{(1,2,1)}_{2,(1,2,1,2)}\mathfrak{S}^{*}_{(1,2,1,2)/(1,1)}\\ &+c^{(1,2,1)}_{2,(1,2,2,1)}\mathfrak{S}^{*}_{(1,2,2,1)/(1,1)}+c^{(1,2,1)}_{2,(1,3,1,1)} \mathfrak{S}^{*}_{(1,3,1,1)/(1,1)}\\&+c^{(1,2,1)}_{2,(2,1,1,2)}\mathfrak{S}^{*}_{(2,1,1,2)/(1,1)}+ c^{(1,2,1)}_{2,(2,1,2,1)}\mathfrak{S}^{*}_{(2,1,2,1)/(1,1)}\\ &+c^{(1,2,1)}_{2,(2,2,1,1)}\mathfrak{S}^{*}_{(2,2,1,1)/(1,1)}+c^{(1,2,1)}_{2,(3,1,1,1)} \mathfrak{S}^{*}_{(3,1,1,1)/(1,1)}\\&+c^{(1,2,1)}_{2,(1,1,4)} \mathfrak{S}^{*}_{(1,1,4)/(1,1)}+c^{(1,2,1)}_{2,(1,2,3)} \mathfrak{S}^{*}_{(1,2,3)/(1,1)}\\&+c^{(1,2,1)}_{2,(1,3,2)} \mathfrak{S}^{*}_{(1,3,2)/(1,1)}+c^{(1,2,1)}_{2,(1,4,1)} \mathfrak{S}^{*}_{(1,4,1)/(1,1)}\\ &+c^{(1,2,1)}_{2,(2,1,3)}\mathfrak{S}^{*}_{(2,1,3)/(1,1)}+ c^{(1,2,1)}_{2,(2,2,2)}\mathfrak{S}^{*}_{(2,2,2)/(1,1)}\\ &+c^{(1,2,1)}_{2,(2,3,1)}\mathfrak{S}^{*}_{(2,3,1)/(1,1)}+c^{(1,2,1)}_{2,(3,1,2)}\mathfrak{S}^{*}_{(3,1,2)/(1,1)}\\ &+c^{(1,2,1)}_{2,(3,2,1)}\mathfrak{S}^{*}_{(3,2,1)/(1,1)}+c^{(1,2,1)}_{2,(4,1,1)}\mathfrak{S}^{*}_{(4,1,1)/(1,1)}. \end{align*} $$

We can compute all the coefficients $c^{\alpha }_{|\beta |-|\alpha |,\beta }$ using Definition 4.1, and most of them turn out to be zero. Hence, we have the following expansion after simplification:

$$ \begin{align*} \begin{aligned} \mathfrak{S}^{*}_{(2)}\cdot\mathfrak{S}^{*}_{(1,2,1)/(1,1)} &=\mathfrak{S}^{*}_{(1,2,1)}-\mathfrak{S}^{*}_{(1,1,2,1)/(1)}-\mathfrak{S}^{*}_{(2,2,1)/(1)}+\mathfrak{S}^{*}_{(2,1,2,1)/ (1,1)}\\ &\quad +{\mathfrak{S}^{*}_{(3,2,1)/(1,1).}} \end{aligned} \end{align*} $$