Some factorizations in the twisted group algebra of symmetric groups

Abstract

In this paper we will give a similar factorization as in [3, 4], where Svrtan and Meljanac examined certain matrix factorizations on Fock-like representation of a multiparametric quon algebra on the free associative algebra of noncommuting polynomials equipped with multiparametric partial derivatives. In order to replace these matrix factorizations (given from the right) by twisted algebra computation, we first consider the natural action of the symmetric group Sn on the polynomial ring Rn in n 2 commuting variables X a b and also introduce a twisted group algebra (defined by the action of Sn on Rn) which we denote by A(Sn). Here we consider some factorizations given from the left because they will be more suitable in calculating the constants (= the elements which are annihilated by all multiparametric partial derivatives) in the free algebra of noncommuting polynomials.

The inverse of a quantum bilinear form of the oriented braid arrangement

“…In [8] we obtained a factorization of certain canonically defined elements in the algebra A(S n ) first as a product of previously defined simpler elements and then as a product of still simpler elements. Now we briefly recall the algebra A(S n ) and some of its canonically defined elements.…”

“…cf. [8]. Of particular interest is its factorization into the product of the simpler elements of the algebra A(S n ).…”

“…Note that the product on the right-hand side of (2.12) is written from right to left for all 1 ≤ k ≤ n − 1. We reproduce here Proposition 3.10 of [8] because it is so important for the further calculation of the inverse matrix of the quantum bilinear form of the oriented braid arrangement. For simplicity, we shall omit the second index n in Proposition 3.10 of [8] when written as Proposition 2.2 below.…”

“…We reproduce here Proposition 3.10 of [8] because it is so important for the further calculation of the inverse matrix of the quantum bilinear form of the oriented braid arrangement. For simplicity, we shall omit the second index n in Proposition 3.10 of [8] when written as Proposition 2.2 below. Let Des(σ) = {1 ≤ i ≤ n − 1 | σ(i) > σ(i + 1)} be the descent set of a permutation σ ∈ S n .…”

“…where we studied the nontrivial factorization of certain canonically defined elements [8]. Furthermore, by using a natural representation of some factorizations of these elements of A(S n ) on the generic weight subspaces B Q of the multiparametric quon algebra B, which is equipped with a multiparametric q-differential structure, we then obtain the corresponding factorizations of the matrix (B * n ) and hence of the matrix (B * n ) −1 , cf.…”