Pieri rule

introduction

• special case of Littlewood-Richardson rule
• expansion of the product of a Schur function with a complete homogeneous symmetric polynomial or an elementary symmetric polynomial
• representation theoretically, it corresponds to a tensor product in which one of the factors is a symmetric or exterior

power of the defining representation

• \(g\)-Pieri is related to complete homogeneous symmetric polynomial
• \(e\)-Pieri is dual to \(g\)-pieri and is related to complete elementary symmetric polynomial
• in more geometric setting, let \(G\) be a classical Lie group and \(P\) a parabolic subgroup. The Pieri rule is a combinatorial formula describing the multiplication by a special Schubert class in the (quantum) cohomology ring of the homogeneous space \(X=G/P\).

Pieri rules for Schur polynomials

• \(S_{\lambda}\) denotes a Schur polynomial of \(k\)-variables

\[ S_{\lambda}S_{(m,0\cdots, 0)}=\sum_{\nu} S_{\nu} \] where the sum is over all \(\nu\) such that \(\nu_1\geq \lambda_1\geq \nu_2 \geq \lambda_2\cdots \geq \nu_k\geq \lambda_k\geq 0\) and \(\sum \nu_j=m+\sum \lambda_j\)

example

• \(S_{(2,1)}S_{(2)}=S_{(4,1)}+S_{(3,2)}+S_{(3,1,1)}+S_{(2,2,1)}\)

generating function form

• recall that \(S_{(m,0\cdots, 0)}=H_m\) and

\[ \prod_{i\geq 1}^k \frac{1}{1-a x_i } = \sum_{j=0}^{\infty}H_ja^j \]

• thus

\[ S_{\mu}\prod_{i\geq 1} \frac{1}{1-a x_i}=\sum_{\lambda\supseteq\mu} a^{\lvert \lambda/\mu \rvert} \varphi_{\lambda/\mu}S_{\lambda} \] where \(\varphi_{\lambda/\mu}=1\) only when \(\lambda/\mu\) is a horizontal strip and zero otherwise

Pieri rules for Macdonal polynomials

• \(g\)- and \(e\)-Pieri rules for Macdonald polynomials expressed in generating function form

\(g\)-Pieri case

\begin{equation} P_{\mu}(q,t) \prod_{i\geq 1} \frac{(atx_i;q)_{\infty}}{(ax_i;q)_{\infty}} =\sum_{\lambda\supseteq\mu} a^{\lvert \lambda/\mu \rvert} \varphi_{\lambda/\mu}(q,t) P_{\lambda}(q,t). \end{equation} Here the Pieri coefficient \(\varphi_{\lambda/\mu}(q,t)=0\) unless \(\lambda/\mu\) is a horizontal strip, in which case \begin{multline}\label{Eq_varphi} \varphi_{\lambda/\mu}(q,t)= \prod_{1\leq i\leq j\leq l(\lambda)} \frac{(qt^{j-i};q)_{\lambda_i-\lambda_j}}{(t^{j-i+1};q)_{\lambda_i-\lambda_j}}\cdot \frac{(qt^{j-i};q)_{\mu_i-\mu_{j+1}}}{(t^{j-i+1};q)_{\mu_i-\mu_{j+1}}} \\ \times \frac{(t^{j-i+1};q)_{\lambda_i-\mu_j}}{(qt^{j-i};q)_{\lambda_i-\mu_j}}\cdot \frac{(t^{j-i+1};q)_{\mu_i-\lambda_{j+1}}}{(qt^{j-i};q)_{\mu_i-\lambda_{j+1}}}. \end{multline}

\(e\)-Pieri case

Similarly, the \(e\)-Pieri rule is given by \begin{equation} P_{\mu}(x;q,t)\prod_{i\geq 1} (1+ax_i)= \sum_{\lambda\supseteq\mu} a^{\lvert \lambda/\mu \rvert} \psi'_{{\lambda}/{\mu}}(q,t) P_{\lambda}(x;q,t), \end{equation} where \(\psi'_{{\lambda}/{\mu}}(q,t)\) is zero unless \(\lambda/\mu\) is a vertical strip, in which case \cite[page 336]{Macdonald95} \begin{equation}\label{Eq_psip} \psi'_{{\lambda}/{\mu}}(q,t) = \prod \frac{1-q^{\mu_i-\mu_j}t^{j-i-1}}{1-q^{\mu_i-\mu_j}t^{j-i}}\cdot \frac{1-q^{\lambda_i-\lambda_j}t^{j-i+1}}{1-q^{\lambda_i-\lambda_j}t^{j-i}}. \end{equation} The product in the above is over all \(i<j\) such that \(\lambda_i=\mu_i\) and \(\lambda_j>\mu_j\). An alternative expression for \(\psi'_{{\lambda}/{\mu}}(q,t)\) is given by \cite[page 340]{Macdonald95} \begin{equation}\label{Eq_psip340} \psi'_{{\lambda}/{\mu}}(q,t)=\prod \frac{b_{\lambda}(s;q,t)}{b_{\mu}(s;q,t)} \end{equation} where the product is over all squares \(s=(i,j)\in\mu\subseteq\lambda\) such that \(i<j\), \(\mu_i=\lambda_i\) and \(\lambda'_j>\mu_j'\).

related items

• Branching rules for Macdonald polynomials

computational resource

• https://drive.google.com/file/d/0B8XXo8Tve1cxekhvdXh2bFctX3M/view

articles

• Soichi Okada, Pieri rules for classical groups and equinumeration between generalized oscillating tableaux and semistandard tableaux, arXiv:1606.02375 [math.CO], June 08 2016, http://arxiv.org/abs/1606.02375

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