Irreducible Monoid Research Articles

Monoid embeddings of symmetric varieties

We determine when an antiinvolution on an adjoint semisimple linear algebraic group extends to an antiinvolution on a $J$-irreducible monoid. Using this information, we study a special class of compactifications of symmetric varieties. Extending the work of Springer on involutions, we describe the parametrizing sets of Borel orbits in these special embeddings.

Picard groups for tropical toric schemes

From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then $Pic(X)$ is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group $CaCl(X_S)$ of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme $X_S$ and prove that $CaCl(X_S)$ is isomorphic to $Pic(X_S)$.

The Orbit Structures of (𝒥, σ)-Irreducible Monoids

Many interesting (𝒥, σ)-irreducible monoids belong to the category of 2-reducible algebraic monoids, i.e., the reductive monoids with exactly two minimal G × G-orbits. Putcha and Renner recently found a combinatorial recipe to determine orbit structures of the 2-reducible algebraic monoids by certain invariants (I +, I −) and (Δ+,Δ−). In this article we give a complete description of the orbit structures of the (𝒥, σ)-irreducible monoids by Putcha and Renner's recipe when σ2 induces the identity on the set of G × G-orbits. This solves all the open problems on finding the orbit structures of (𝒥, σ)-irreducible monoids of type (n ≥ 4) and .

Orders of the Renner monoids

In this paper, we give a general formula for a Renner monoid of a reductive monoid with zero. It is based on Putcha's type map. We apply this formula to find the orders of the Renner monoids for J -irreducible monoids K ∗ ρ ( G ) ¯ where G is a simple algebraic group over an algebraically closed field K, and ρ : G → GL ( V ) is an irreducible representation associated with an arbitrary fundamental dominant weight.

The Lattice of [formula omitted]-Classes of ([formula omitted], σ)-Irreducible Monoids, II

A reductive monoid is an algebraic monoid with a reductive unit group. Generally, we have the question of finding the orbits of the unit group of a reductive monoid acting on both sides of the monoid. Putcha and Renner give a recipe to determine the orbits for J-irreducible monoids according to the Dynkin diagrams. We obtain a similar recipe for the question to (J,σ)-irreducible monoids (not J-irreducible) of type D2n. However, there is no similar answer for types An (n≥4) and E26. The fixed points of any (J,σ)-irreducible monoid under σ is a finite reductive monoid. We obtain that any such finite reductive monoid is J-irreducible. Then we find the orbits of these monoids under the two sided action of their unit groups.

Types of Reductive Monoids

Let M be a reductive monoid with a reductive unit group G. Clearly there is a natural G×G action on M. The orbits are the J-classes (in the sense of semigroup theory) and form a finite lattice. The general problem of finding the lattice remains open. In this paper we study a new class of reductive monoids constructed by multilined closure. We obtain a general theorem to determine the lattices of these monoids. We find that the (J,σ)-irreducible monoids of Suzuki type and Ree type belong to this new class. Using the general theorem we then list all the lattices and type maps of the (J,σ)-irreducible monoids of Suzuki type and Ree type.

The classification of (J, σ)-irreducible monoids of type A 4

In this paper we develop a very basic method to classify (J, σ)-irreducible monoids of type A 4. As a typical example, we list all the types for the monoids corresponding to the strongly dominant weights. This example also shows that there is no general theorem to determine the cross-section lattices for reductive monoids according to their Dynkin diagrams as Putcha and Renner’s recipe for J-irreducible monoids.

The Lattice of [formula omitted]-Classes of ([formula omitted], σ)-Irreducible Monoids

In this paper, we introduce a new concept, namely, the (J,σ)-irreducible monoid. LetG0be a simple algebraic group. Let σ be a surjective endomorphism ofG0such that (G0)σ, the fixed points ofG0under σ, is finite. We construct a (J,σ)-irreducible monoidMwithGthe unit group. Extending σ toM, we obtain a finite monoidMσwhich is J-irreducible. IfMis J-irreducible, we give a precise recipe to determine the lattices of J-classes ofMandMσ.

Toral subgroups lying in the centralizer of the group of units

Let S be a compact connected finite dimensional monoid whose group of units G is a compact connected Lie group. Then there is an open set W about the unit element such that any compact subgroup within W has dimension at most dim ⁡ S − dim ⁡ G − 1 \dim S - \dim G - 1 and if any toral subgroup achieves this dimension then that toral subgroup lies in the centralizer of G. Two applications are given, one to embeddings of irreducible monoids into S.