The Worpitzky identity for the groups of signed and even-signed permutations

Seminar

Speaker

Eli Bagno (Jerusalem College of Technology)

Date

06/12/2020 - 15:30 - 14:00Add to Calendar 2020-12-06 14:00:00 2020-12-06 15:30:00 The Worpitzky identity for the groups of signed and even-signed permutations The well-known Worpitzky identity provides a connection between two bases of : The standard basis and the binomial basis , where the Eulerian numbers for the Coxeter group of type (the symmetric group) serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types and (signed and even-signed permutations groups, respectively) using generatingfunctionology. Motivated by Foata-Schützenberger and Rawlings' proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs of this identity and for its -analogue in the Coxeter groups of types and . Our proofs utilize the language of -partitions for the and posets introduced by Chow and Stembridge, respectively. Joint work with David Garber and Mordechai Novick. Zoom אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public

Place

Zoom

Abstract

The well-known Worpitzky identity provides a connection between two bases of $\mathbb{Q}[x]$ : The standard basis $(x+1)^n$ and the binomial basis ${{x+n-i} \choose {n}}$ , where the Eulerian numbers for the Coxeter group of type $A$ (the symmetric group) serve as the entries of the transformation matrix.

Brenti has generalized this identity to the Coxeter groups of types $B$ and $D$ (signed and even-signed permutations groups, respectively) using generatingfunctionology.

Motivated by Foata-Schützenberger and Rawlings' proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs of this identity and for its $q$ -analogue in the Coxeter groups of types $B$ and $D$ . Our proofs utilize the language of $P$ -partitions for the $B_n$ and $D_n$ posets introduced by Chow and Stembridge, respectively.

Joint work with David Garber and Mordechai Novick.

Last Updated Date : 01/12/2020