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.