Odd length in Weyl groups

Abstract

We define a new statistic on any Weyl group which we call the odd length and which reduces, for Weyl groups of types A, B, and D, the the statistics by the same name that have already been defined and studied in [10], [13], [14], and [3]. We show that the signed (by length) generating function of the odd length always factors nicely except possibly in type E 8 , and we obtain multivariate analogues of these factorizations in types B and D.

Odd and Even Major Indices and One-Dimensional Characters for Classical Weyl Groups

“…Similar computations show that no additive "odd major index" that depends only on the descent and negative sets exists in the hyperoctahedral groups that is equidistributed with the odd length, where the odd length is the one defined in [30] and [31], and further studied in [10,12], and [22], namely L B (σ) = 1 2 |{(i, j) ∈ [±n] 2 : i < j, σ(i) > σ(j), i ≡ j (mod 2)}|, where σ ∈ B n and σ(0) := 0. Analogous considerations about more general indexes, as at the end of the previous paragraph, also apply in this case.…”

“…In recent years, a new statistic on the symmetric groups has been introduced and studied in relation with vector spaces over finite fields equipped with a certain quadratic form [21]. This statistic combines combinatorial and parity conditions and is now known as the odd inversion number, or odd length [10,12]. Analogous statistics have later been defined and studied for the hyperoctahedral and even hyperoctahedral groups [11,30,31], and more recently for all Weyl groups [12].…”

“…This statistic combines combinatorial and parity conditions and is now known as the odd inversion number, or odd length [10,12]. Analogous statistics have later been defined and studied for the hyperoctahedral and even hyperoctahedral groups [11,30,31], and more recently for all Weyl groups [12]. A crucial property of this new statistic is that its signed (by length) generating function over the corresponding Weyl group always factors explicitly [12,33].…”

Odd diagrams, Bruhat order, and pattern avoidance

“…Bruhat intervals arising as odd diagram classes are not, however, self-dual in general. For example, if D = {(1, 1), (1,2), (1,3), (1,5), (2,4), (3,1), (3,2), (3,3), (5, 2), (5, 3), (7, 3)} and n = 9 then Perm 9 (D) = [654172839, 958172634] and one can check that this interval is not self-dual. like to thank Tobias Rossmann for help with some computations.…”

“…Odd analogues of well-known combinatorial objects and statistics associated with permutations (and, more generally, with Weyl and Coxeter group elements) have been recently considered and studied (see, for instance, [3,4,5,6,7,10,13,14,15]). In particular, odd analogues of permutation diagrams, called odd diagrams, were introduced and studied in [6].…”