"Pieri rule"의 두 판 사이의 차이
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| + | ==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 | ||
| + | |||
| + | [[분류:symmetric polynomials]] | ||
| + | |||
| + | == 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 | ||
| + | [[분류:migrate]] | ||
2020년 11월 13일 (금) 02:19 판
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 $\lambda'_j>\mu_j'$.
computational resource
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