Journal of Algebra 228, 497–511 (2000) doi:10.1006/jabr.1999.8278, available online at http://www.idealibrary.com on The Eigenvalue Distribution of a Random Unipotent Matrix in Its Representation on Lines Jason Fulman View metadata, citationDepartment and similar of Mathematics,papers at core.ac.uk Stanford University, Building 380, MC 2125, brought to you by CORE Stanford, California 94305 provided by Elsevier - Publisher Connector E-mail: [email protected] Communicated by Walter Feit Received July 13, 1999 The eigenvalue distribution of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed arc of the unit circle and offer an approach to other asymptotics. For the case of all unipotent matrices, the proof gives a probabilis- tic interpretation to identities of Macdonald from symmetric function theory. For the case of upper triangular matrices over a finite field, connections between sym- metric function theory and a probabilistic growth algorithm of Borodin and Kirillov emerge. © 2000 Academic Press Key Words: random matrix; symmetric functions; Hall–Littlewood polynomial. 1. INTRODUCTION The subject of eigenvalues of random matrices is very rich. The eigen- value spacings of a complex unitary matrix chosen from Haar measure re- late to the spacings between the zeros of the Riemann zeta function [O, RS1, RS2]. For further recent work on random complex unitary matrices, see [DiS, R, So, W]. The references [Dy] and [Me] contain much of inter- est concerning the eigenvalues of a random matrix chosen from Dyson’s or- thogonal, unitary, and symplectic circular ensembles, for instance, connec- tions with the statistics of nuclear energy levels. The papers [AD, BaiDeJ, Ok] and the references contained in them give exciting recent results relat- ing eigenvalue distributions of matrices to statistics of random permutations such as the longest increasing subsequence. 497 0021-8693/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved. 498 jason fulman Little work seems to have been done on the eigenvalue statistics of ele- ments chosen from finite groups. One recent step is Chapter 5 of Wieand’s thesis [W]. She studies the permutation eigenvalues of a random element of the symmetric group in its representation on the set 1;:::;n. This note gives two natural q-analogs of Wieand’s work. For the first q-analog, let α 2 GLn; q be a random unipotent matrix. Letting V be the vector space on which α acts, we consider the eigenvalues of α in the permutation representation of GLn; q on the lines of V .LetXθα be the number of eigenvalues of α lying in a fixed arc 1;ei2πθ; 0 <θ<1, of the unit circle. Bounds are obtained for the mean of Xθ (we suspect that as n !1with q fixed, a normal limit theorem holds). A second q-analog which we analyze is the case when α is a randomly chosen unipotent upper triangular matrix over a finite field. A third interesting q-analog would be taking α uniformly chosen in GLn; q; however, this seems intractable. It would also be of interest to extend Wieand’s work to more general representations of the symmetric group, using formulas of Stembridge [Ste]. The main method of this paper is to interpret identities of symmetric function theory in a probabilistic setting. Section 2 gives background and results in this direction. This interaction appears fruitful, and it is shown for instance that a probabilistic algorithm of Borodin and Kirillov describing the Jordan form of a random unipotent upper triangular matrix [Bo, Ki1] follows from the combinatorics of symmetric functions. This ties in with work on analogous algorithms for the unipotent conjugacy classes of finite classical groups [F1]. The applications to the eigenvalue problems described above appear in Section 3. We remark that quite different computations in symmetric function theory play the central role in the work of Diaconis and Shahshahani [DiS] on the eigenvalues of random complex classical matrices. 2. SYMMETRIC FUNCTIONS To begin we describe some notation, asP on pages 2–5 of [Ma]. Let λ be a partition of a non-negative integer n D i λi into non-negative integral parts λ1 ≥ λ2 ≥···≥0. The notation λ=n will mean that λ is a partition 0 of n.Letmiλ be the number of parts of λ of size i, and let λ be the partition dual to λ in the sense that λ0 D m λ+m λC···.Letnλ be P i i iC1 the quantity i≥1i − 1λi. It is also useful to define the diagram associated 2 to λ as the set of points i; j∈Z such that 1 ≤ j ≤ λi. We use the convention that the row index i increases as one goes downward and the column index j increases as one goes across. So the diagram of the partition the eigenvalue distribution 499 5441 is ::::: :::: :::: : L λi Let Gλ be an abelian p-group isomorphic to i Cycp . We write G D λ if G is an abelian p-group isomorphic to G . Finally, let λ 1 1 1 D 1 − ··· 1 − : r p r p p The rest of the paper will treat the case GLn; p with p prime as op- posed to GLn; q. This reduction is made only to make the paper more accessible in places, allowing us to use the language of abelian p-groups rather than modules over power series rings. From Chapter 2 of Macdon- ald [Ma] it is clear that everything works for prime powers. 2.1. Unipotent Elements of GLn; p It is well known that the unipotent conjugacy classes of GLn; p are parametrized by partitions λ of n. A representative of the class λ is given by M 000 λ1 0 Mλ 00 2 ; 00Mλ ··· 3 000··· where Mi is the i ∗ i matrix of the form 110··· ··· 0 0110··· 0 0011··· 0 : ··· ··· ··· ··· ··· ··· ··· ··· ··· 011 000··· 01 500 jason fulman Lemmas 1–3 recall elementary facts about unipotent elements in GLn; p. Lemma 1. [Ma, p. 181; SS]. The number of unipotent elements in GLn; p with conjugacy class type λ is GLn; p P Q : λ0 2 1 p i i p miλ Chapter 3 of [Ma] defines Hall–Littlewood symmetric functions Pλx1;x2;:::y t which will be used extensively. There is an explicit formula for the Hall–Littlewood polynomials. Let the permutation w act on the x-variables by sending xi to xwi. There is also a coordinate-wise λ action of w on λ =λ1;:::;λn and Sn is defined as the subgroup of Sn stabilizing λ in this action. For a partition λ =λ1;:::;λn of length ≤ n, two formulas for the Hall–Littlewood polynomial restricted to n variables are: " # ! 1 X Y x − tx λ1 λn i j P x ;:::;x y t= w x ···xn λ 1 n Q Qm λ r 1 i 1−t xi − xj i≥0 rD1 1−t w2Sn i<j X λ λ Y xi − txj D w x 1 ···x n 1 n x − x λ λ >λ i j w2Sn=Sn i j Lemma 2. The probability that a unipotent element of GLn; p has con- jugacy class of type λ is equal to either of n 1 p p n 1. P 0 2 Q p λi 1 i p miλ pn 1 P 1 ; 1 ; 1 ;···y 1 2. p n λ p p2 p3 p pnλ Proof. The first statement follows from Lemma 1 and Steinberg’s theo- rem that GLn; p has pnn−1 unipotent elements. The second statement follows from the first and from elementary manipulations applied to Mac- donald’s principal specialization formula (page 337 of [Ma]). Full details appear in [F2]. One consequence of Lemma 2 is that in the p !1limit, all mass is placed on the partition λ =n. Thus the asymptotics in this paper will focus on the more interesting case of the fixed p, n !1limit. Lemma 3. X 1 1 P Q D λ0 2 1 1 p i pn λ`n i p miλ p n the eigenvalue distribution 501 Proof. Immediate from Lemma 2. Lemmas 4 and 5 relate to the theory of Hall polynomials and Hall– Littlewood symmetric functions [Ma]. Lemma 4, for instance, is the duality property of Hall polynomials. Lemma 4. [Ma, p. 181.] For all partitions λ; µ; ν, G1 ⊆ Gλ x Gλ=G1 D µ; G1 D ν=G1 ⊆ Gλ x Gλ=G1 D ν; G1 D µ: Lemma 5. Let Gλ denote an abelian p-group of type λ and G1 a subgroup. Then for all types µ, X G1 ⊆ Gλ x G1 D µ 1 1 P Q D P Q : λ0 2 1 µ0 2 1 1 p i p i pn−µ λ`n i p miλ i p miµ p n−µ Proof. Macdonald (page 220 of [Ma]), using Hall–Littlewood symmetric functions, establishes for any partitions µ; ν the equation X G1 ⊆ Gλ x Gλ=G1 D µ; G1 D ν P Q λ0 2 1 p i λxλ=µ+ν i p miλ 1 1 D P Q P Q : µ0 2 1 ν02 1 p i p i i p miµ i p miν Fixing µ, summing the left-hand side over all ν of size n −µ, and applying Lemma 4 yields X X G1 ⊆ Gλ x Gλ=G1 D µ; G1 D ν P Q λ0 2 1 p i λ ν i p miλ X X G1 ⊆ Gλ x Gλ=G1 D ν; G1 D µ D P Q λ0 2 1 p i λ ν i p miλ X G1 ⊆ Gλ x G1 D µ D P Q : λ0 2 1 p i λ i p miλ Fixing µ, summing the right-hand side over all ν of size n −µ, and apply- ing Lemma 3 gives that 1 X 1 P Q P Q µ0 2 1 ν02 1 p i p i i p miµ ν`n−µ i p miν 1 1 D P Q ; µ0 2 1 1 p i pn−µ i p miµ p n−µ proving the lemma.