A -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

DOI:
https://doi.org/10.1090/S0002-9947-98-01880-7

MathSciNet review:
1407491

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system . Let be the Varchenko matrix for this arrangement with all hyperplane parameters equal to . We show that is the matrix with rows and columns indexed by permutations with entry equal to where is the number of inversions of . Equivalently is the matrix for left multiplication on by

Clearly commutes with the right-regular action of on . A general theorem of Varchenko applied in this special case shows that is singular exactly when is a root of for some between and . In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the -module structure of the nullspace of in the case that is singular. Our first result is that

in the case that where Lie denotes the multilinear part of the free Lie algebra with generators. Our second result gives an elegant formula for the determinant of restricted to the virtual -module with characteristic the power sum symmetric function .

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Additional Information

**Phil Hanlon**

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

Email:
phil@math.lda.umich.edu

**Richard P. Stanley**

Affiliation:
Department of Mathematics 2-375, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
rstan@math.mit.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-01880-7

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