Igusa-type functions associated to finite formed spaces and their functional equations

Abstract

Abstract. We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, Hermitian or quadratic form. To this end we introduce Igusa-type rational functions encoding these polynomials and prove that they satisfy certain functional equations.Some of our results are achieved by expressing the polynomials in question in terms of what we call parabolic length functions on Coxeter groups of type A. While our treatment of the orthogonal case exploits combinatorial properties of integer compositions and their refinements, we formulate a precise conjecture how in this situation, too, the polynomials may be described in terms of parabolic length functions.

Odd length for even hyperoctahedral groups and signed generating functions

“…The following statistic was first defined in [9]. Our definition is not the original one, but is equivalent to it (see [9, Definition 5.1 and Lemma 5.2]) and is the one that is best suited for our purposes.…”

“…In the next section we recall some definitions, notation, and results that are used in the sequel. In §3 we define a new statistic on the even hyperoctahedral group which is a natural analogue of the odd length statistics that have already been defined in types A and B in [9] and [18], and study some general properties of the corresponding signed generating functions. These include a complementation property, the identification of subsets of the quotients over which the corresponding signed generating function always vanishes, and operations on a quotient that leave the corresponding signed generating function unchanged.…”

Odd length in Weyl groups

“…This statistic depends on the root system underlying the Weyl group and we compute it combinatorially for the classical root systems of types A, B, C, and D. As a consequence we verify that this statistic coincides, in types A, B, and D, with the odd length statistics defined and studied in [10], [13], [14], [3], [5], and [11] in these types. Our combinatorial computation of the statistic in the classical types shows that it is the sum of some more fundamental statistics and we compute the signed (by length) multivariate generating function of these statistics in types B and D. These results reduce to results in [10], [14], and [2] when all the variables are equal. We also show that the signed generating function of this statistic factors nicely for any crystallographic root system except possibly in type E 8 .…”

“…This statistic depends on the choice of a simple system in the root system of the Weyl group and we show that its generating function over the Weyl group only depends on the root system. Using a convenient choice of simple system we compute combinatorially the odd length of any element of any Weyl group of classical type and verify that it coincides, in types A, B, and D, with the odd length statistics already defined in [10], [13], [14], [3], and [5], in these types. In §4 we show that the signed generating function of the odd length over the symmetric group coincides with the one over the unimodal permutations.…”

“…A new statistic on the symmetric group was introduced in [10] in relation to formed spaces. This statistic combines combinatorial and parity conditions and is now known as the odd length.…”