Generalized Burnside algebra of type B_{n}

Yıl 2020, Cilt: 69 Sayı: 1, 252 - 265, 30.06.2020

Hasan Arslan Himmet Can

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

Cited By: 1

Öz

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 $W_{n}$.

Anahtar Kelimeler

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

Kaynakça

• Bergeron, F., Bergeron, N., Howlett, R.B., Taylor, D.E., A Decomposition of the Descent Algebra of a Finite Coxeter Group, Journal of Algebraic Combinatorics, 1(1) 1992, 23--44.
• Bonnafé, C., Hohlweg, C., Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups, Ann. Inst. Fourier (Grenoble) 56(1) 2006, 131--181.
• Bonnafé, C., Representation theory of Mantaci-Reutenauer algebras, Algebras and Representation Theory 11(4) (2008), 307--346.
• Carter, R.W., Conjugacy Classes in the Weyl Groups, Compositio Math., 25 1972, 1--59.
• Curtis, C.W., Reiner, I., Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. II, John Wiley and Sons, 1987.
• Douglass, J.M., Pfeiffer, G., Rohrle, G., On reflection subgroups of finite Coxeter groups, Comm. Algebra 41(7) 2013, 2574--2592.
• Fleischmann, P., On pointwise conjugacy of distinguished coset representatives in Coxeter groups, J. Group Theory, 5 (2002), 269--283.
• Geck, M., Pfeiffer, G., Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Mathematical Society Monographs, New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000.
• Humphreys, J.E., Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Math., vol. 29, Cambridge University Press, 1990.
• Mantaci, R., Reutenauer, C., A generalization of Solomon's algebra for hyperoctahedral groups and other wreath products, Comm. Algebra, 23(1) (1995), 27--56.
• Solomon, L., A Mackey formula in the group ring of a Coxeter group, J. Algebra, 41(2) (1976), 255--264.