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Generalized Burnside algebra of type B_{n}

Year 2020, Volume: 69 Issue: 1, 252 - 265, 30.06.2020
https://doi.org/10.31801/cfsuasmas.599246

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 $W_{n}$.

References

  • 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.
Year 2020, Volume: 69 Issue: 1, 252 - 265, 30.06.2020
https://doi.org/10.31801/cfsuasmas.599246

Abstract

References

  • 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.
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Hasan Arslan 0000-0002-0430-8737

Himmet Can This is me 0000-0001-8485-6815

Publication Date June 30, 2020
Submission Date July 31, 2019
Acceptance Date October 18, 2019
Published in Issue Year 2020 Volume: 69 Issue: 1

Cite

APA Arslan, H., & Can, H. (2020). Generalized Burnside algebra of type B_{n}. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1), 252-265. https://doi.org/10.31801/cfsuasmas.599246
AMA Arslan H, Can H. Generalized Burnside algebra of type B_{n}. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2020;69(1):252-265. doi:10.31801/cfsuasmas.599246
Chicago Arslan, Hasan, and Himmet Can. “Generalized Burnside Algebra of Type B_{n}”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69, no. 1 (June 2020): 252-65. https://doi.org/10.31801/cfsuasmas.599246.
EndNote Arslan H, Can H (June 1, 2020) Generalized Burnside algebra of type B_{n}. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 1 252–265.
IEEE H. Arslan and H. Can, “Generalized Burnside algebra of type B_{n}”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 69, no. 1, pp. 252–265, 2020, doi: 10.31801/cfsuasmas.599246.
ISNAD Arslan, Hasan - Can, Himmet. “Generalized Burnside Algebra of Type B_{n}”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69/1 (June 2020), 252-265. https://doi.org/10.31801/cfsuasmas.599246.
JAMA Arslan H, Can H. Generalized Burnside algebra of type B_{n}. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69:252–265.
MLA Arslan, Hasan and Himmet Can. “Generalized Burnside Algebra of Type B_{n}”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 69, no. 1, 2020, pp. 252-65, doi:10.31801/cfsuasmas.599246.
Vancouver Arslan H, Can H. Generalized Burnside algebra of type B_{n}. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69(1):252-65.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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