A deformation of a trivial symmetric group action
Authors:
Phil Hanlon and Richard P. Stanley
Journal:
Trans. Amer. Math. Soc. 350 (1998), 44454459
MSC (1991):
Primary 20C30, 05E10
MathSciNet review:
1407491
Fulltext PDF Free Access
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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 rightregular 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 481091109
Email:
phil@math.lda.umich.edu
Richard P. Stanley
Affiliation:
Department of Mathematics 2375, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email:
rstan@math.mit.edu
DOI:
http://dx.doi.org/10.1090/S0002994798018807
PII:
S 00029947(98)018807
Received by editor(s):
June 25, 1996
Additional Notes:
The first author was partially supported by the National Science Foundation, Grant No. DMS9500979
The second author was partially supported by the National Science Foundation, Grant No. DMS9206374.
Article copyright:
© Copyright 1998
American Mathematical Society
