Pieri rule - 편집 역사

2021년 2월 17일 (수) 10:02에 Pythagoras0님의 편집

2021-02-17T10:02:06Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T08:08:05Z

메타데이터: 새 문단

2020년 11월 16일 (월) 18:02에 Pythagoras0님의 편집

2020-11-16T18:02:44Z

2020년 11월 16일 (월) 17:50에 Pythagoras0님의 편집

2020-11-16T17:50:13Z

2020년 11월 13일 (금) 10:19에 imported>Pythagoras0님의 편집

2020-11-13T10:19:47Z

2020년 11월 13일 (금) 10:19에 imported>Pythagoras0님의 편집

2020-11-13T10:19:47Z

imported>Pythagoras0: /* introduction */

2017-06-12T06:38:46Z

introduction

2017년 6월 6일 (화) 02:19에 imported>Pythagoras0님의 편집

2017-06-06T02:19:01Z

imported>Pythagoras0: /* generating function form */

2017-06-05T06:17:35Z

generating function form

2017년 6월 5일 (월) 05:47에 imported>Pythagoras0님의 편집

2017-06-05T05:47:08Z

@@ 92번째 줄: / 92번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q7191884 Q7191884]

← 이전 판		2020년 12월 28일 (월) 08:08 판
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]

@@ 4번째 줄: / 4번째 줄: @@
 * 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
+* <math>g</math>-Pieri is related to complete homogeneous symmetric polynomial
-* $e$-Pieri is dual to $g$-pieri and is related to complete elementary 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 $G$ 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 $X=G/P$.
+* 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==
-* $S_{\lambda}$ denotes a Schur polynomial of $k$-variables
+* <math>S_{\lambda}</math> denotes a Schur polynomial of <math>k</math>-variables
-$$
+:<math>
 S_{\lambda}S_{(m,0\cdots, 0)}=\sum_{\nu} S_{\nu}
-$$
+</math>
-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 $\sum \nu_j=m+\sum \lambda_j$
+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===
-* $S_{(2,1)}S_{(2)}=S_{(4,1)}+S_{(3,2)}+S_{(3,1,1)}+S_{(2,2,1)}$
+* <math>S_{(2,1)}S_{(2)}=S_{(4,1)}+S_{(3,2)}+S_{(3,1,1)}+S_{(2,2,1)}</math>
 ===generating function form===
-* recall that $S_{(m,0\cdots, 0)}=H_m$ and
+* recall that <math>S_{(m,0\cdots, 0)}=H_m</math> and
-$$
+:<math>
 \prod_{i\geq 1}^k \frac{1}{1-a x_i } = \sum_{j=0}^{\infty}H_ja^j
-$$
+</math>
 * thus
-$$
+:<math>
 S_{\mu}\prod_{i\geq 1} \frac{1}{1-a x_i}=\sum_{\lambda\supseteq\mu} a^{\lvert \lambda/\mu \rvert} \varphi_{\lambda/\mu}S_{\lambda}
-$$
+</math>
-where $\varphi_{\lambda/\mu}=1$ only when $\lambda/\mu$ is a horizontal strip and zero otherwise
+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==
-* $g$- and $e$-Pieri rules for Macdonald polynomials expressed in generating function form
+* <math>g</math>- and <math>e</math>-Pieri rules for Macdonald polynomials expressed in generating function form
-===$g$-Pieri case===
+===<math>g</math>-Pieri case===
 \begin{equation}
 P_{\mu}(q,t) \prod_{i\geq 1} \frac{(atx_i;q)_{\infty}}{(ax_i;q)_{\infty}}
@@ 40번째 줄: / 40번째 줄: @@
 \end{equation}
 Here the Pieri coefficient
-$\varphi_{\lambda/\mu}(q,t)=0$ unless $\lambda/\mu$ is a horizontal strip,
+<math>\varphi_{\lambda/\mu}(q,t)=0</math> unless <math>\lambda/\mu</math> is a horizontal strip,
 in which case
 \begin{multline}\label{Eq_varphi}
@@ 53번째 줄: / 53번째 줄: @@
-===$e$-Pieri case===
+===<math>e</math>-Pieri case===
-Similarly, the $e$-Pieri rule is given by
+Similarly, the <math>e</math>-Pieri rule is given by
 \begin{equation}
 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),
 \end{equation}
-where $\psi'_{{\lambda}/{\mu}}(q,t)$ is zero
+where <math>\psi'_{{\lambda}/{\mu}}(q,t)</math> is zero
-unless $\lambda/\mu$ is a vertical strip, in which case
+unless <math>\lambda/\mu</math> is a vertical strip, in which case
 \cite[page 336]{Macdonald95}
 \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}}.
 \end{equation}
-The product in the above is over all $i<j$ such that
+The product in the above is over all <math>i<j</math> such that
-$\lambda_i=\mu_i$ and $\lambda_j>\mu_j$.
+<math>\lambda_i=\mu_i</math> and <math>\lambda_j>\mu_j</math>.
-An alternative expression for $\psi'_{{\lambda}/{\mu}}(q,t)$ is given by
+An alternative expression for <math>\psi'_{{\lambda}/{\mu}}(q,t)</math> 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 $s=(i,j)\in\mu\subseteq\lambda$
+where the product is over all squares <math>s=(i,j)\in\mu\subseteq\lambda</math>
-such that $i<j$, $\mu_i=\lambda_i$ and $\lambda'_j>\mu_j'$.
+such that <math>i<j</math>, <math>\mu_i=\lambda_i</math> and <math>\lambda'_j>\mu_j'</math>.
 ==related items==

← 이전 판	2020년 11월 13일 (금) 10:19 판
1번째 줄:	1번째 줄:
	+	==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]]

← 이전 판		2020년 11월 13일 (금) 10:19 판
1번째 줄:		1번째 줄:
−	~~==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

← 이전 판	2020년 11월 16일 (월) 17:50 판
(차이 없음)

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
 * special case of [[Littlewood-Richardson rule]]
-* expansion of
+* expansion of the product of a Schur function with a complete homogeneous symmetric polynomial or an elementary symmetric polynomial
 * $g$-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==