Characters of Renner monoids and their Hecke algebras

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type [Formula: see text] Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type [Formula: see text] and use them to obtain an explicit description of the character table for the type [Formula: see text] Renner monoid Hecke algebra.

ROOT SEMIGROUPS IN REDUCTIVE MONOIDS

It is well known that in a reductive group, the Borel subgroup is a product of the maximal torus and the one-dimensional positive root subgroups. The purpose of this paper is to find an analog of this result for reductive monoids. Via a study of reductive monoids of semisimple rank 1, we introduce the concept of root semigroups. By analyzing the associated root elements in the Renner monoid, we show that the closure of the Borel subgroup is generated by the maximal torus and positive root semigroups. Along the way we generalize the Jordan decomposition of algebraic groups to reductive monoids.

R-POLYNOMIALS OF FINITE MONOIDS OF LIE TYPE

This paper studies the combinatorics of the orbit Hecke algebras associated with W × W orbits in the Renner monoid of a finite monoid of Lie type, M, where W is the Weyl group associated with M. It is shown by Putcha in [12] that the Kazhdan–Lusztig involution [6] can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using the R-polynomials, we calculate the Möbius function of the Bruhat–Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.

PRESENTATION FOR RENNER MONOIDS

We extend the result obtained in E. Godelle [‘The braid rook monoid’, Internat. J. Algebra Comput.18 (2008), 779–802] to every Renner monoid: we provide a monoid presentation for Renner monoids, and we introduce a length function which extends the Coxeter length function and which behaves nicely.

REPRESENTATIONS OF THE RENNER MONOID

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalize the Munn–Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn–Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown.

THE RENNER MONOIDS AND CELL DECOMPOSITIONS OF THE SYMPLECTIC ALGEBRAIC MONOIDS

In this paper we explicitly determine the Renner monoid ℛ and the cross section lattice Λ of the symplectic algebraic monoid MSpn in terms of the Weyl group and the concept of admissible sets; it turns out that ℛ is a submonoid of ℛn, the Renner monoid of the whole matrix monoid Mn, and that Λ is a sublattice of Λn, the cross section lattice of Mn. Cell decompositions in algebraic geometry are usually obtained by the method of [1]. We give a more direct definition of cells for MSpn in terms of the B × B-orbits, where B is a Borel subgroup of the unit group G of MSpn. Each cell turns out to be the intersection of MSpn with a cell of Mn. We also show how to obtain these cells using a carefully chosen one parameter subgroup.