Generalized Burnside algebra of type B_{n}

Year 2020, Volume: 69 Issue: 1, 252 - 265, 30.06.2020

Hasan Arslan , Himmet Can

https://doi.org/10.31801/cfsuasmas.599246

Cited By: 1

Abstract

In this paper, we firstly give an alternative method to determine the size of $C(S_{n})$ which is the set of elements of type $S_{n}$ in a finite Coxeter system $(W_{n},S_{n})$ of type $B_{n}$. We also show that all cuspidal classes of $W_{n}$ are actually the conjugate classes $\mathcal{K}_{\lambda}$ for every $\lambda \in \mathcal{DP}^{+}(n)$. We then define the generalized Burnside algebra $HB(W_{n})$ for $W_{n}$ and construct a surjective algebra morphism between $HB(W_{n})$ and Mantaci-Reutenauer algebra $\mathcal{MR}(W_{n})$. We obtain a set of orthogonal primitive idempotents $e_{\lambda}$, $\lambda \in \mathcal{DP}(n)$ of $HB(W_{n})$, that is, all the characteristic class functions of $W_{n}$. Finally, we give an effective formula to compute the number of elements of all the conjugate classes $\mathcal{K}_{\lambda}$, $\lambda \in \mathcal{DP}(n)$ of $W_{n}$.

Keywords

Cuspidal Class, Mantaci-Reutenauer Algebra, Burnside Algebra, Orthogonal Primitive Idempotents

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