Signed Generating Functions Of Odd Length For The Even Hyperoctahedral Groups

The odd length on Weyl groups is a new statistic analogous to the classical Coxeter length function,and features parity conditions.As a natural generalization of the odd length on Weyl group of types A and B,Francesco Brenti and Angela Carnevale defined an analogue statistic on the even hyperoctahedral groups(i.e.,Weyl groups of type D).Then they derived explicit closed product formulas for the signed generating functions of this statistic over the whole group and over its maximal and some other quotients.Based on the results that have been obtained,they proposed three conjectures about the formulas for the signed generating functions of the even signed hyperoctahedral groups.We establish explicit closed product formulas for the signed generating functions of the odd length over arbitrary parabolic quotients of the even hyperoctahedral groups.The main approach is to use sign reversing involutions to construct appropriate supporting sets for the signed generating functions,and then we use parabolic factorizations to decompose the signed generating functions into products of well known polynomials.As consequences,we verify the three conjectures posed by Francesco Brenti and Angela Carnevale.We also give necessary and sufficient conditions for the signed generating functions to have expressions as products of cyclotomic polynomials,settling a conjecture of John R.Stembridge that some signed generating functions of the even hyperoctahedral groups cannot be expressed as products of cyclotomic polynomials.