"Hall algebra"의 두 판 사이의 차이
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| + | ==introduction== | ||
| + | <math> | ||
| + | \newcommand{\la}{\lambda} | ||
| + | \newcommand{\La}{\Lambda} | ||
| + | </math> | ||
| + | |||
| + | ==Hall polynomials== | ||
| + | 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 <math>s_{\mu}</math> is the Schur function. | ||
| + | |||
| + | 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}. | ||
| + | \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 <math>c^{\la}_{\mu \nu} = 0</math>, then <math>g^{\la}_{\mu \nu}(t) = 0</math> as a function of <math>t</math>. | ||
| + | * 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 <math>g^{\la}_{\mu \nu}(t) = g^{\la}_{\nu \mu}(t)</math>. | ||
| + | |||
| + | |||
| + | 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 <math>f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})</math>. | ||
| + | |||
| + | |||
| + | ==related items== | ||
| + | * {{수학노트|url=홀-리틀우드(Hall-Littlewood)_대칭함수}} | ||
| + | |||
| + | |||
==expositions== | ==expositions== | ||
* Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940. | * Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940. | ||
| 5번째 줄: | 44번째 줄: | ||
==articles== | ==articles== | ||
* Scherotzke, Sarah, and Nicolo Sibilla. “Quiver Varieties and Hall Algebras.” arXiv:1506.03609 [math], June 11, 2015. http://arxiv.org/abs/1506.03609. | * 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'}] | ||
2021년 2월 17일 (수) 01:39 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 \(f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})\).
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'}]