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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A $\lowercase{q}$-deformation of
a trivial symmetric group action

Authors: Phil Hanlon and Richard P. Stanley
Journal: Trans. Amer. Math. Soc. 350 (1998), 4445-4459
MSC (1991): Primary 20C30, 05E10
MathSciNet review: 1407491
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Abstract: Let $\mathcal{A}$ be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system $A_{n-1}$. Let $B=B(q)$ be the Varchenko matrix for this arrangement with all hyperplane parameters equal to $q$. We show that $B$ is the matrix with rows and columns indexed by permutations with $\sigma, \tau$ entry equal to $q^{i(\sigma \tau^{-1})}$ where $i(\pi)$ is the number of inversions of $\pi$. Equivalently $B$ is the matrix for left multiplication on $\mathbb{C}\mathfrak{S}_n$ by

\begin{displaymath}\Gamma _n(q)=\sum _{\pi \in \mathfrak{S}_n} q^{i(\pi)} \pi . \end{displaymath}

Clearly $B$ commutes with the right-regular action of $\mathfrak{S}_n$ on $\mathbb{C}\mathfrak{S}_n$. A general theorem of Varchenko applied in this special case shows that $B$ is singular exactly when $q$ is a $j(j-1)^{st}$ root of $1$ for some $j$ between $2$ and $n$. In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the $\mathfrak{S}_n$-module structure of the nullspace of $B$ in the case that $B$ is singular. Our first result is that

\begin{displaymath}\ker(B) = \mathrm{ind}^{\mathfrak{S}_n}_{\mathfrak{S}_{n-1}} (\mathrm{Lie}_{n-1}) /\mathrm{Lie}_n\end{displaymath}

in the case that $q = e^{2\pi i/n(n-1)}$ where Lie$_n$ denotes the multilinear part of the free Lie algebra with $n$ generators. Our second result gives an elegant formula for the determinant of $B$ restricted to the virtual $\mathfrak{S}_n$-module $P^\eta$ with characteristic the power sum symmetric function $p_\eta(x)$.

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Additional Information

Phil Hanlon
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Richard P. Stanley
Affiliation: Department of Mathematics 2-375, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): June 25, 1996
Additional Notes: The first author was partially supported by the National Science Foundation, Grant No. DMS-9500979
The second author was partially supported by the National Science Foundation, Grant No. DMS-9206374.
Article copyright: © Copyright 1998 American Mathematical Society