Absolute order in general linear groups

Abstract

Abstract. This paper studies a partial order on the general linear group GL(V ) called the absolute order, derived from viewing GL(V ) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V ) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including self-duality of its intervals.Working over a finite field F q , it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL n (F q ) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.

GLn(Fq)-analogues of factorization problems in the s

“…This phenomenon is mysterious. A similar phenomenon was observed in the discussion following Theorem 4.2 in [HLR16], namely, that the counting formula q e(α) (q n − 1) k−1 for factorizations of a regular elliptic element in GL n (F q ) into k factors with fixed space codimensions given by the composition α of n is a q-analogue of the counting formula n k−1 for factorizations of an n-cycle as a genus-0 product of k cycles of specified lengths. On the other hand, we can give a heuristic explanation for the fact that the lower indices in the M -coefficients in Theorem 1.4 are shifted by 1 compared with those in Theorem 1.3: the matrix group S n does not act irreducibly in its standard representation, as every permutation fixes the all-ones vector.…”

“…Recently, there has been interest in q-analogues of such problems, replacing S n with the finite general linear group GL n (F q ), the long cycle with a Singer cycle (or, more generally, regular elliptic element) c, and the number of cycles with the fixed space dimension [LRS14,HLR16]; or in more general geometric settings [HLRV11]. In the present paper, we extend this approach to give the following q-analogue of Theorem 1.1.…”

“…In GL n (F q ), the necessary character theory was worked out by Green [Gre55], building on work of Steinberg [Ste51]. This approach has been used recently by the first-named author and coauthors to count factorizations of Singer cycles into reflections [LRS14] and to count genus-0 factorizations (that is, those in which the codimensions of the fixed spaces of the factors sum to the codimension of the fixed space of the product) of regular elliptic elements [HLR16]. The current work subsumes these previous results while requiring no new character values.…”

Stability of the centers of group algebras of GL(q)

“…As observed in [HLR17], a basic property about the reflection length of g ∈ GL n (q) is that it coincides with the codimension of its fixed point subspace in F n q . We show that the reflection length of an element g ∈ GL n (q) is equal to the size of its modified type.…”

“…The combinatorics of partial orders on G n arising from the reflection lengths has been studied in [HLR17]. Recall the codimension codimV g n = n − dim V g n = rank(g − I n ).…”