Generalized Burnside algebra of type B_{n}

Authors : Hasan ARSLAN, Himmet CAN

Pages : 252-265

Doi:10.31801/cfsuasmas.599246

View : 40 | Download : 11

Publication Date : 2020-06-30

Article Type : Research Paper

Abstract :In this paper, we firstly give an alternative method to determine the size of $Cinsert ignore into journalissuearticles values(S_{n});$ which is the set of elements of type $S_{n}$ in a finite Coxeter system $insert ignore into journalissuearticles values(W_{n},S_{n});$ of type $B_{n}$. We also show that all cuspidal classes of $W_{n}$ are actually the conjugate classes $\mathcal{K}_{\lambda}$ for every $\lambda \in \mathcal{DP}^{+}insert ignore into journalissuearticles values(n);$. We then define the generalized Burnside algebra $HBinsert ignore into journalissuearticles values(W_{n});$ for $W_{n}$ and construct a surjective algebra morphism between $HBinsert ignore into journalissuearticles values(W_{n});$ and Mantaci-Reutenauer algebra $\mathcal{MR}insert ignore into journalissuearticles values(W_{n});$. We obtain a set of orthogonal primitive idempotents $e_{\lambda}$, $\lambda \in \mathcal{DP}insert ignore into journalissuearticles values(n);$ of $HBinsert ignore into journalissuearticles values(W_{n});$, 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 $\mathcal{K}_{\lambda}$, $\lambda \in \mathcal{DP}insert ignore into journalissuearticles values(n);$ of $W_{n}$.
Keywords : Cuspidal Class, Mantaci Reutenauer Algebra, Burnside Algebra, Orthogonal Primitive Idempotents

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