"Pieri rule"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| (같은 사용자의 중간 판 2개는 보이지 않습니다) | |||
| 4번째 줄: | 4번째 줄: | ||
* representation theoretically, it corresponds to a tensor product in which one of the factors is a symmetric or exterior | * representation theoretically, it corresponds to a tensor product in which one of the factors is a symmetric or exterior | ||
power of the defining representation | power of the defining representation | ||
| − | * | + | * <math>g</math>-Pieri is related to complete homogeneous symmetric polynomial |
| − | * | + | * <math>e</math>-Pieri is dual to <math>g</math>-pieri and is related to complete elementary symmetric polynomial |
| − | * in more geometric setting, let | + | * in more geometric setting, let <math>G</math> be a classical Lie group and <math>P</math> 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 <math>X=G/P</math>. |
==Pieri rules for Schur polynomials== | ==Pieri rules for Schur polynomials== | ||
| − | * | + | * <math>S_{\lambda}</math> denotes a Schur polynomial of <math>k</math>-variables |
| − | + | :<math> | |
S_{\lambda}S_{(m,0\cdots, 0)}=\sum_{\nu} S_{\nu} | S_{\lambda}S_{(m,0\cdots, 0)}=\sum_{\nu} S_{\nu} | ||
| − | + | </math> | |
| − | where the sum is over all | + | where the sum is over all <math>\nu</math> such that <math>\nu_1\geq \lambda_1\geq \nu_2 \geq \lambda_2\cdots \geq \nu_k\geq \lambda_k\geq 0</math> and <math>\sum \nu_j=m+\sum \lambda_j</math> |
===example=== | ===example=== | ||
| − | * | + | * <math>S_{(2,1)}S_{(2)}=S_{(4,1)}+S_{(3,2)}+S_{(3,1,1)}+S_{(2,2,1)}</math> |
===generating function form=== | ===generating function form=== | ||
| − | * recall that | + | * recall that <math>S_{(m,0\cdots, 0)}=H_m</math> and |
| − | + | :<math> | |
\prod_{i\geq 1}^k \frac{1}{1-a x_i } = \sum_{j=0}^{\infty}H_ja^j | \prod_{i\geq 1}^k \frac{1}{1-a x_i } = \sum_{j=0}^{\infty}H_ja^j | ||
| − | + | </math> | |
* thus | * thus | ||
| − | + | :<math> | |
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} | 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} | ||
| − | + | </math> | |
| − | where | + | where <math>\varphi_{\lambda/\mu}=1</math> only when <math>\lambda/\mu</math> is a horizontal strip and zero otherwise |
==Pieri rules for Macdonal polynomials== | ==Pieri rules for Macdonal polynomials== | ||
| − | * | + | * <math>g</math>- and <math>e</math>-Pieri rules for Macdonald polynomials expressed in generating function form |
| − | === | + | ===<math>g</math>-Pieri case=== |
\begin{equation} | \begin{equation} | ||
P_{\mu}(q,t) \prod_{i\geq 1} \frac{(atx_i;q)_{\infty}}{(ax_i;q)_{\infty}} | P_{\mu}(q,t) \prod_{i\geq 1} \frac{(atx_i;q)_{\infty}}{(ax_i;q)_{\infty}} | ||
| 40번째 줄: | 40번째 줄: | ||
\end{equation} | \end{equation} | ||
Here the Pieri coefficient | Here the Pieri coefficient | ||
| − | + | <math>\varphi_{\lambda/\mu}(q,t)=0</math> unless <math>\lambda/\mu</math> is a horizontal strip, | |
in which case | in which case | ||
\begin{multline}\label{Eq_varphi} | \begin{multline}\label{Eq_varphi} | ||
| 53번째 줄: | 53번째 줄: | ||
| − | === | + | ===<math>e</math>-Pieri case=== |
| − | Similarly, the | + | Similarly, the <math>e</math>-Pieri rule is given by |
\begin{equation} | \begin{equation} | ||
P_{\mu}(x;q,t)\prod_{i\geq 1} (1+ax_i)= | P_{\mu}(x;q,t)\prod_{i\geq 1} (1+ax_i)= | ||
| 60번째 줄: | 60번째 줄: | ||
\psi'_{{\lambda}/{\mu}}(q,t) P_{\lambda}(x;q,t), | \psi'_{{\lambda}/{\mu}}(q,t) P_{\lambda}(x;q,t), | ||
\end{equation} | \end{equation} | ||
| − | where | + | where <math>\psi'_{{\lambda}/{\mu}}(q,t)</math> is zero |
| − | unless | + | unless <math>\lambda/\mu</math> is a vertical strip, in which case |
\cite[page 336]{Macdonald95} | \cite[page 336]{Macdonald95} | ||
\begin{equation}\label{Eq_psip} | \begin{equation}\label{Eq_psip} | ||
| 68번째 줄: | 68번째 줄: | ||
\frac{1-q^{\lambda_i-\lambda_j}t^{j-i+1}}{1-q^{\lambda_i-\lambda_j}t^{j-i}}. | \frac{1-q^{\lambda_i-\lambda_j}t^{j-i+1}}{1-q^{\lambda_i-\lambda_j}t^{j-i}}. | ||
\end{equation} | \end{equation} | ||
| − | The product in the above is over all | + | The product in the above is over all <math>i<j</math> such that |
| − | + | <math>\lambda_i=\mu_i</math> and <math>\lambda_j>\mu_j</math>. | |
| − | An alternative expression for | + | An alternative expression for <math>\psi'_{{\lambda}/{\mu}}(q,t)</math> is given by |
\cite[page 340]{Macdonald95} | \cite[page 340]{Macdonald95} | ||
\begin{equation}\label{Eq_psip340} | \begin{equation}\label{Eq_psip340} | ||
\psi'_{{\lambda}/{\mu}}(q,t)=\prod \frac{b_{\lambda}(s;q,t)}{b_{\mu}(s;q,t)} | \psi'_{{\lambda}/{\mu}}(q,t)=\prod \frac{b_{\lambda}(s;q,t)}{b_{\mu}(s;q,t)} | ||
\end{equation} | \end{equation} | ||
| − | where the product is over all squares | + | where the product is over all squares <math>s=(i,j)\in\mu\subseteq\lambda</math> |
| − | such that | + | such that <math>i<j</math>, <math>\mu_i=\lambda_i</math> and <math>\lambda'_j>\mu_j'</math>. |
==related items== | ==related items== | ||
| 91번째 줄: | 91번째 줄: | ||
* 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 | * 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]] | [[분류:migrate]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q7191884 Q7191884] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'pieri'}, {'LOWER': "'s"}, {'LEMMA': 'formula'}] | ||
2021년 2월 17일 (수) 02:02 기준 최신판
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
메타데이터
위키데이터
- ID : Q7191884
Spacy 패턴 목록
- [{'LOWER': 'pieri'}, {'LOWER': "'s"}, {'LEMMA': 'formula'}]