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 .

**[B]**L.C. Biedenharn,*J. Phys. A*(1989), L873-L878. MR**90k:17027****[BV]**T. Brylawski and A. Varchenko, ``The determinant formula for a matroid bilinear form", preprint.**[DKKT]**G. Duchamp, A.A. Klyachko, D. Krob, J.Y. Thibon, ``Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras", preprint.**[F]**W. Feit,*Characters of Finite Groups*, Yale University Press, New Haven, Conn., 1965.**[G]**A. Garsia, ``*Combinatorics of the free Lie algebra and the symmetric group"*, Analysis etcetera, J. Moser's festschrift, Academic Press, New York (1990). MR**91a:17006****[Gr]**O.W. Greenberg, ``*Example of infinite statistics*", Phys. Rev. Lett.,**64**(1990), 705-708. MR**90k:81080****[M]**I.G. Macdonald,*Symmetric Functions and Hall Polynomials*, Oxford University Press, 1979. MR**84g:05001****[O]**P. Orlik,*Introduction to arrangements*, CBMS Regional Series in Mathematics, #72, AMS, Providence, RI (1989). MR**90i:32018****[R1]**C. Reutenauer,*Free Lie Algebras*, Oxford University Press, 1993.**[R2]**C. Reutenauer, ``*Theorem of Poincaré-Birkhoff-Witt and symmetric group representations of degrees equal to the Stirling numbers"*, Lecture Notes in Math., vol. 1234, Springer-Verlag, N.Y. (1986). MR**89i:05029****[Ro]**A. Robinson, ``*The space of fully grown trees*", preprint.**[Ste]**J. Stembridge, ``*On the eigenvalues of representations of reflection groups and wreath products*", Pacific J. Math.,**140**, No. 2 (1989), 353-396. MR**91a:20022****[SV]**V. Schechtman and A. Varchenko, ``*Arrangements of hyperplanes and Lie algebra homology*" Inventiones Math.,**106**(1991), 139-194. MR**93b:17067****[V]**A. Varchenko, ``*Bilinear form of real configuration of hyperplanes*'', Adv. in Math.**97**(1993), 110-144. MR**94b:52023****[W]**S. Whitehouse, ``*Gamma homology of commutative algebras and some related representations of the symmetric group*", preprint.**[Z]**D. Zagier, ``*Realizability of a model in infinite statistics*", Comm. in Math. Phys.,**147**(1992), 199-210. MR**93i:81105**

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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