A 𝑞-deformation of a trivial symmetric group action

Hanlon, Phil; Stanley, Richard

doi:10.1090/S0002-9947-98-01880-7

A $q$-deformation of a trivial symmetric group action
HTML articles powered by AMS MathViewer

by Phil Hanlon and Richard P. Stanley PDF

Trans. Amer. Math. Soc. 350 (1998), 4445-4459 Request permission

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 \[ \Gamma _n(q)=\sum _{\pi \in \mathfrak {S}_n} q^{i(\pi )} \pi . \] 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 \[ \ker (B) = \mathrm {ind}^{\mathfrak {S}_n}_{\mathfrak {S}_{n-1}} (\mathrm {Lie}_{n-1}) /\mathrm {Lie}_n\] 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 $p_\eta (x)$.

References

L. C. Biedenharn, The quantum group $\textrm {SU}_q(2)$ and a $q$-analogue of the boson operators, J. Phys. A 22 (1989), no. 18, L873–L878. MR 1015226
T. Brylawski and A. Varchenko, “The determinant formula for a matroid bilinear form", preprint.
G. Duchamp, A.A. Klyachko, D. Krob, J.Y. Thibon, “Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras", preprint.
W. Feit, Characters of Finite Groups, Yale University Press, New Haven, Conn., 1965.
Adriano M. Garsia, Combinatorics of the free Lie algebra and the symmetric group, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 309–382. MR 1039352
O. W. Greenberg, Example of infinite statistics, Phys. Rev. Lett. 64 (1990), no. 7, 705–708. MR 1036450, DOI 10.1103/PhysRevLett.64.705
D. Frank Hsu, Cyclic neofields and combinatorial designs, Lecture Notes in Mathematics, vol. 824, Springer-Verlag, Berlin-New York, 1980. MR 616639
Peter Orlik, Introduction to arrangements, CBMS Regional Conference Series in Mathematics, vol. 72, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. MR 1006880, DOI 10.1090/cbms/072
C. Reutenauer, Free Lie Algebras, Oxford University Press, 1993.
Christophe Reutenauer, Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985) Lecture Notes in Math., vol. 1234, Springer, Berlin, 1986, pp. 267–284. MR 927769, DOI 10.1007/BFb0072520
A. Robinson, “The space of fully grown trees", preprint.
John R. Stembridge, On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140 (1989), no. 2, 353–396. MR 1023791
Vadim V. Schechtman and Alexander N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), no. 1, 139–194. MR 1123378, DOI 10.1007/BF01243909
Alexandre Varchenko, Bilinear form of real configuration of hyperplanes, Adv. Math. 97 (1993), no. 1, 110–144. MR 1200291, DOI 10.1006/aima.1993.1003
S. Whitehouse, “Gamma homology of commutative algebras and some related representations of the symmetric group", preprint.
Don Zagier, Realizability of a model in infinite statistics, Comm. Math. Phys. 147 (1992), no. 1, 199–210. MR 1171767