A Note About the Trace Functions on Mantaci--Reutenauer Algebra

Year 2018, Volume: 6 Issue: 1, 140 - 143, 15.04.2018

Hasan Arslan

Abstract

In this paper, we obtain the trace functions of Mantaci-Reutenauer algebra $\mathcal{MR}(W_{n})$, where $(W_{n},S_{n})$ is a Coxeter system of type $B_{n}$. We also show for every $\lambda \in \mathcal{DP}(n)$ that each characteristic class function $e_{\lambda}$ of the group $W_{n}$ is a trace function of Mantaci-Reutenauer algebra, where $\mathcal{DP}(n)$ stands for the set of all double partitions of $n$. Since the dimension of the trace function space on the Mantaci-Reutenauer algebra is $|\mathcal{DP}(n)|$, it exactly coincides with the algebra $\mathbb{Q}\textrm{Irr}W_{n}$ generated by the irreducible characters of the group $W_{n}$. Although the multiplication of basis elements $d_{A}$ and $d_{A'}$ is not commutative in Mantaci-Reutenauer algebra, the images of $d_{A}d_{A'}$ and $d_{A'}d_{A}$ under $e_{\lambda}$ are equal to each other.

Keywords

Mantaci-Reutenauer algebra, Orthogonal primitive idempotent, Trace function

References

• [1] Arslan, H., Descent Algebras of Bn-type Coxeter Groups, PhD Thesis, Erciyes University, 2015.
• [2] C. Bonnaf´e and C. Hohlweg, Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups, Ann. Inst. Fourier (Grenoble) Volume:56 No.1 (2006), 131-181.
• [3] C. Bonnaf´e, Representation theory of Mantaci-Reutenauer algebras, Algebr. Represent. Theory Volume: 11, No.4 (2008), 307-346.
• [4] M. Geck, Trace functions on Iwahori-Hecke algebras, Knot theory (Warsaw, 1995), Banach Center Publ., Polish Acad. Sci., Warsaw Volume:42 (1998), 87–109.
• [5] Geck, M. and 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.
• [6] R. Mantaci and C. Reutenauer A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products, Comm. Algebra Volume:23, No.1 (1995), 27-56.
• [7] L. Solomon, A Mackey formula in the group ring of a Coxeter group, Journal of Algebra Volume:41, No.2 (1976), 255-264.