Mathematics Home / Research Seminars: Algebra and Combinatorics

Time & Location: All talks are on Wednesday in Gibson Hall 127 at 3:00 PM unless otherwise noted.

Organizer: Mahir Can

**Title:** Decomposable Specht modules

**Liron Speyer | Okinawa Institute of Science and Technology**

**Abstract:**

I will give a brief survey of the study of decomposable Specht modules for the symmetric group and its Hecke algebra, which includes results of Murphy, Dodge and Fayers, and myself, before reporting on recent work with Louise Sutton, in which we have studied decomposable Specht modules for the Hecke algebra of type $B$ indexed by `bihooks’. I will present our conjectured classification of decomposable Specht modules indexed by bihooks, which we proved `one half of’, and some ongoing work in explicitly determining the structure of those decomposable Specht modules.

**Location:** Gibson Hall 127

**Time: **3:00

**Title:** *Gorenstein polytopes*

**Takayuki Hibi | Osaka University**

**Abstract:**

A Gorenstein polytope is a lattice polytope one of whose dilated polytopes is a reflexive polytope. In my talk, after reviewing Gorenstein polytopes from a viewpoint of enumeration of lattice points, several conjectures arising from Gorenstein polytopes will be reported. No special knowledge will be required to understand my talk.

**Title:** *Total nonnegativity and induced sign characters of the Hecke algebra*

**Mark Skandera | Lehigh University**

**Abstract:**

Gantmacher's study of totally nonnegative (TNN) matrices in the 1930's eventually found applications in many areas of mathematics. Descending from his work are problems concerning TNN polynomials, those polynomial functions of n^2 variables which take nonnegative values on TNN matrices. Closely related to TNN polynomials are functions in the Hecke algebra trace space whose evaluations at certain Hecke algebra elements yield polynomials in N[q]. In all cases, it would be desirable to combinatorially interpret the resulting nonnegative numbers. In 2017, Kaliszewski, Lambright, and the presenter found the first cancellation-free combinatorial formula for the evaluation of all elements of a basis of V at all elements of a basis of the Hecke algebra. We will discuss a recent improvement upon this result which also advances our understanding of TNN polynomials. This is joint work with Adam Clearwater.

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**Title:** *Symbolic powers and the (stable) containment problem*

**Eloísa Grifo | University of California at Riverside**

**Abstract:**

The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals, which arise naturally from the theory of primary decomposition, are difficult to compute but have a natural geometric description.

In trying to compare symbolic and ordinary powers, Harbourne conjectured that a famous containment by Ein--Lazersfeld--Smith, Hochster--Huneke, and Ma--Schwede could be improved. Harbourne's Conjecture is a statement depending on n that has been disproved for particular values of n. However, recent evidence points towards a stable version of Harbourne's conjecture, where we substitute all n by all n large enough. Some of that evidence is joint work with Craig Huneke and Vivek Mukundan.

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**Title:** *Classical Mechanics, Symplectic Geometry, Combinatorics*

**Tewodros Amdeberhan | Tulane University**

**Abstract:**

In this semi-expository talk, we give a brief on classical mechanics in the Hamiltonian setting, describe it in the symplectic framework and draw out some interesting combinatorics. The discussion will be accessible to all.

**Title:** *The Borel Submonoid of a Symplectic Monoid*

**Mahir Can | Tulane University**

**Abstract:**

The symplectic monoid MSp_n of this talk is the semisimple monoid that is obtained from the defining representation of the symplectic group Sp_n. The Zariski closure of a Borel subgroup of MSp_n is called a Borel submonoid of MSp_n.

In this talk, we will discuss some geometric and combinatorial properties of the symplectic Borel submonoids.On the combinatorial side, we will show that there is a new type BC set partition combinatorics associated with these objects.

On the geometric side, we will show that such Borel submonoids are rationally smooth varieties. Consequently, our new set partitions can be used for computing the intersection cohomology Betti numbers of the symplectic Borel submonoids. This is a joint work with Hayden Houser and Corey Wolfe.

**Title:** *Some multiplicity one theorems for wreath products*

**Yiyang She - Tulane University**

**Abstract:**

Let G be a group and let H be a subgroup.

If all irreducible representations of G restrict to multiplicity free H representations, then (G,H) is said to be a strong Gelfand pair.

In this talk we will present our recent results on the strong Gelfand pairs of finite wreath products.

This is a joint work with Mahir Bilen Can and Liron Speyer.

**Title:** *Topological Group Algebras and Topological Universal Enveloping Algebras for Lie Algebras*

**Karl H. Hofmann | Technische Universitaet Darmstadt**

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