Hall algebra

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.

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