"Hall algebra"의 두 판 사이의 차이

2021년 2월 17일 (수) 02:39 기준 최신판

introduction

\( \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \)

Hall polynomials

Recall that the Littlewood-Richardson coefficient \(c^{\la}_{\mu \nu}\) is equal to the number of tableaux \(T\) of shape \(\la - \mu\) and weight \(\nu\) such that \(w(T)\), the word of \(T\), is a lattice permutation. We have \begin{equation*} s_{\mu}s_{\nu} = \sum_{\la} c^{\la}_{\mu \nu} s_{\la}, \end{equation*} where \(s_{\mu}\) is the Schur function.

We briefly recall the Hall polynomials \(g_{\mu \nu}^{\la}(q)\) \cite[Chs. II and V]{Mac}. Let \(\mathcal{O}\) be a complete (commutative) discrete valuation ring, \(\mathcal{P}\) its maximal ideal and \(k = \mathcal{O}/\mathcal{P}\) the residue field. We assume \(k\) is a finite field. Let \(q\) be the number of elements in \(k\). Let \(M\) be a finite \(\mathcal{O}\)-module of type \(\la\). Then the number of submodules of \(N\) of \(M\) with type \(\nu\) and cotype \(\mu\) is a polynomial in \(q\), called the Hall polynomial, denoted \(g_{\mu \nu}^{\la}(q)\). One can consider our motivating case of \(\mathbb{Q}_{p}\) and its ring of integers \(\mathcal{O} = \mathbb{Z}_{p}\) and \(G = Gl_{n}(\mathbb{Q}_{p})\), so that \(q=p\). Then they are also the structure constants for the ring \(\mathcal{H}(G^{+},K)\). That is, for \(\mu, \nu \in \La_{2n}^{+}\), we have \begin{equation*} c_{\mu} \star c_{\nu} = \sum_{\la \in \La_{2n}^{+}} g_{\mu \nu}^{\la}(p) c_{\la}. \end{equation*} Note that, in particular, \begin{equation*} g_{\mu \nu}^{\la}(p) = (c_{\mu} \star c_{\nu})(p^{\la}) = \int_{G} c_{\mu}(p^{\la}y^{-1})c_{\nu}(y)dy = meas.(p^{\la}Kp^{-\nu}K \cap Kp^{\mu}K). \end{equation*}

Several important facts are known (see \cite[Ch. II]{Mac}):

If \(c^{\la}_{\mu \nu} = 0\), then \(g^{\la}_{\mu \nu}(t) = 0\) as a function of \(t\).
If \(c^{\la}_{\mu \nu} \neq 0\), then \(g^{\la}_{\mu \nu}(t)\) has degree \(n(\la) - n(\mu) - n(\nu)\) and leading coefficient \(c^{\la}_{\mu \nu}\), where the notation \(n(\la) = \sum (i-1) \la_{i}\).
We have \(g^{\la}_{\mu \nu}(t) = g^{\la}_{\nu \mu}(t)\).

Also if one multiplies two Hall-Littlewood polynomials, and expands the result in the Hall-Littlewood basis, one has \begin{equation*} P_{\mu}(x;t) P_{\nu}(x;t) = \sum_{\la} f^{\la}_{\mu \nu}(t) P_{\la}(x;t), \end{equation*} with \(f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})\).

related items

틀:수학노트

expositions

Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940.

articles

Scherotzke, Sarah, and Nicolo Sibilla. “Quiver Varieties and Hall Algebras.” arXiv:1506.03609 [math], June 11, 2015. http://arxiv.org/abs/1506.03609.

메타데이터

위키데이터

ID : Q5642657

Spacy 패턴 목록

[{'LOWER': 'hall'}, {'LEMMA': 'algebra'}]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-$
+<math>
 \newcommand{\la}{\lambda}
 \newcommand{\La}{\Lambda}
-$
+</math>
 ==Hall polynomials==
-Recall that the Littlewood-Richardson coefficient $c^{\la}_{\mu \nu}$ is equal to the number of tableaux $T$ of shape $\la - \mu$ and weight $\nu$ such that $w(T)$, the word of $T$, is a lattice permutation.  We have
+Recall that the Littlewood-Richardson coefficient <math>c^{\la}_{\mu \nu}</math> is equal to the number of tableaux <math>T</math> of shape <math>\la - \mu</math> and weight <math>\nu</math> such that <math>w(T)</math>, the word of <math>T</math>, is a lattice permutation.  We have
 \begin{equation*}
 s_{\mu}s_{\nu} = \sum_{\la} c^{\la}_{\mu \nu} s_{\la},
 \end{equation*}
-where $s_{\mu}$ is the Schur function.
+where <math>s_{\mu}</math> is the Schur function.
-We briefly recall the Hall polynomials $g_{\mu \nu}^{\la}(q)$ \cite[Chs. II and V]{Mac}.  Let $\mathcal{O}$ be a complete (commutative) discrete valuation ring, $\mathcal{P}$ its maximal ideal and $k = \mathcal{O}/\mathcal{P}$ the residue field.  We assume $k$ is a finite field.  Let $q$ be the number of elements in $k$.  Let $M$ be a finite $\mathcal{O}$-module of type $\la$.  Then the number of submodules of $N$ of $M$ with type $\nu$ and cotype $\mu$ is a polynomial in $q$, called the Hall polynomial, denoted $g_{\mu \nu}^{\la}(q)$.  One can consider our motivating case of $\mathbb{Q}_{p}$ and its ring of integers $\mathcal{O} = \mathbb{Z}_{p}$ and $G = Gl_{n}(\mathbb{Q}_{p})$, so that $q=p$.  Then they are also the structure constants for the ring $\mathcal{H}(G^{+},K)$.  That is, for $\mu, \nu \in \La_{2n}^{+}$, we have
+We briefly recall the Hall polynomials <math>g_{\mu \nu}^{\la}(q)</math> \cite[Chs. II and V]{Mac}.  Let <math>\mathcal{O}</math> be a complete (commutative) discrete valuation ring, <math>\mathcal{P}</math> its maximal ideal and <math>k = \mathcal{O}/\mathcal{P}</math> the residue field.  We assume <math>k</math> is a finite field.  Let <math>q</math> be the number of elements in <math>k</math>.  Let <math>M</math> be a finite <math>\mathcal{O}</math>-module of type <math>\la</math>.  Then the number of submodules of <math>N</math> of <math>M</math> with type <math>\nu</math> and cotype <math>\mu</math> is a polynomial in <math>q</math>, called the Hall polynomial, denoted <math>g_{\mu \nu}^{\la}(q)</math>.  One can consider our motivating case of <math>\mathbb{Q}_{p}</math> and its ring of integers <math>\mathcal{O} = \mathbb{Z}_{p}</math> and <math>G = Gl_{n}(\mathbb{Q}_{p})</math>, so that <math>q=p</math>.  Then they are also the structure constants for the ring <math>\mathcal{H}(G^{+},K)</math>.  That is, for <math>\mu, \nu \in \La_{2n}^{+}</math>, we have
 \begin{equation*}
 c_{\mu} \star c_{\nu} = \sum_{\la \in \La_{2n}^{+}} g_{\mu \nu}^{\la}(p) c_{\la}.
@@ 22번째 줄: / 22번째 줄: @@
 Several important facts are known (see \cite[Ch. II]{Mac}):
-* If $c^{\la}_{\mu \nu} = 0$, then $g^{\la}_{\mu \nu}(t) = 0$ as a function of $t$.
+* If <math>c^{\la}_{\mu \nu} = 0</math>, then <math>g^{\la}_{\mu \nu}(t) = 0</math> as a function of <math>t</math>.
-* If $c^{\la}_{\mu \nu} \neq 0$, then $g^{\la}_{\mu \nu}(t)$ has degree $n(\la) - n(\mu) - n(\nu)$ and leading coefficient $c^{\la}_{\mu \nu}$, where the notation $n(\la) = \sum (i-1) \la_{i}$.
+* If <math>c^{\la}_{\mu \nu} \neq 0</math>, then <math>g^{\la}_{\mu \nu}(t)</math> has degree <math>n(\la) - n(\mu) - n(\nu)</math> and leading coefficient <math>c^{\la}_{\mu \nu}</math>, where the notation <math>n(\la) = \sum (i-1) \la_{i}</math>.
-* We have $g^{\la}_{\mu \nu}(t) = g^{\la}_{\nu \mu}(t)$.
+* We have <math>g^{\la}_{\mu \nu}(t) = g^{\la}_{\nu \mu}(t)</math>.
@@ 31번째 줄: / 31번째 줄: @@
 P_{\mu}(x;t) P_{\nu}(x;t) = \sum_{\la} f^{\la}_{\mu \nu}(t) P_{\la}(x;t),
 \end{equation*}
-with $f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})$.
+with <math>f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})</math>.
@@ 44번째 줄: / 44번째 줄: @@
 ==articles==
 * Scherotzke, Sarah, and Nicolo Sibilla. “Quiver Varieties and Hall Algebras.” arXiv:1506.03609 [math], June 11, 2015. http://arxiv.org/abs/1506.03609.
+[[분류:migrate]]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q5642657 Q5642657]
+===Spacy 패턴 목록===
+* [{'LOWER': 'hall'}, {'LEMMA': 'algebra'}]