"Hall algebra"의 두 판 사이의 차이
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | + | <math> | |
\newcommand{\la}{\lambda} | \newcommand{\la}{\lambda} | ||
\newcommand{\La}{\Lambda} | \newcommand{\La}{\Lambda} | ||
| − | + | </math> | |
==Hall polynomials== | ==Hall polynomials== | ||
| − | Recall that the Littlewood-Richardson coefficient | + | 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*} | \begin{equation*} | ||
s_{\mu}s_{\nu} = \sum_{\la} c^{\la}_{\mu \nu} s_{\la}, | s_{\mu}s_{\nu} = \sum_{\la} c^{\la}_{\mu \nu} s_{\la}, | ||
\end{equation*} | \end{equation*} | ||
| − | where | + | where <math>s_{\mu}</math> is the Schur function. |
| − | We briefly recall the Hall polynomials | + | 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*} | \begin{equation*} | ||
c_{\mu} \star c_{\nu} = \sum_{\la \in \La_{2n}^{+}} g_{\mu \nu}^{\la}(p) c_{\la}. | 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}): | Several important facts are known (see \cite[Ch. II]{Mac}): | ||
| − | * If | + | * 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 | + | * 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 | + | * 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), | P_{\mu}(x;t) P_{\nu}(x;t) = \sum_{\la} f^{\la}_{\mu \nu}(t) P_{\la}(x;t), | ||
\end{equation*} | \end{equation*} | ||
| − | with | + | with <math>f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})</math>. |
2020년 11월 16일 (월) 04:30 판
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.