arXiv subjects

You'an Cao

Publications and source records attributed to You'an Cao.

4 recordsLinked to original sources

arXiv2015

Cross-section Lattices of ${\mathcal J}$-irreducible Monoids and Orbit Structures of Weight Polytopes

Let $λ$ be a dominant weight of a finite dimensional simple Lie algebra and $W$ the Weyl group. The convex hull of $Wλ$ is defined as the weight polytope of $λ$. We provide a new proof that there is a natural bijection between the set of orbits of the nonempty faces of the weight polytope under the action of the Weyl group and the set of the connected subdiagrams of the extended Dynkin diagram that contain the extended node $\{-λ\}$. We show that each face of the polytope can be transformed to a standard parabolic face. We also show that a standard parabolic face is the convex hull of the orbit of a parabolic subgroup of $W$ acting on the dominant weight. In addition, we find that the linear space spanned by a face is in fact spanned by roots.

math.RT↗

arXiv2012

Conjugacy Classes of Renner Monoids

In this paper we describe conjugacy classes of a Renner monoid $R$ with unit group $W$, the Weyl group. We show that every element in $R$ is conjugate to an element $ue$ where $u\in W$ and $e$ is an idempotent in a cross section lattice. Denote by $W(e)$ and $W_*(e)$ the centralizer and stabilizer of $e\in Λ$ in $W$, respectively. Let $W(e)$ act by conjugation on the set of left cosets of $W_*(e)$ in $W$. We find that $ue$ and $ve$ ($u, v\in W$) are conjugate if and only if $uW_*(e)$ and $vW_*(e)$ are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugacy classes of $R$ and the orbits of this action. We then obtain a formula for calculating the number of conjugacy classes of $R$, and describe in detail the conjugacy classes of the Renner monoid of some $\cal J$-irreducible monoids. We then generalize the Munn conjugacy on a rook monoid to any Renner monoid and show that the Munn conjugacy coincides with the semigroup conjugacy, action conjugacy, and character conjugacy. We also show that the number of inequivalent irreducible representations of $R$ over an algebraically closed field of characteristic zero equals the number of the Munn conjugacy classes in $R$.

math.RT↗

arXiv2008

Orders of Finite Reductive Monoids

We show four formulas for calculating the orders of finite reductive monoids with zero. As applications, these formulas are then used to calculate the orders of finite reductive monoids induced from the $F_q$-split $\J$-irreducible monoids $\overline {K^*ρ(G_0)}$ where $G_0$ is a simple algebraic group over the algebraic closure of $F_q$, and $ρ: G_0\to GL(V)$ is the irreducible representation associated with any dominant weight. Finally, we give an explicit formula for the orders of finite symplectic monoids associated with the last fundamental dominant weight of type $C_l$; the connections to $H$-polynomials and Betti numbers are shown.

math.GR↗

arXiv2008

Representations of the Renner Monoid

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalizes 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.

math.RT↗