## Algebra Seminar at UJ

### Summer semester 2020

#### Abstracts

**Maciej Dołęga - A Positive Combinatorial Formula for Symplectic Kostka-Foulkes Polynomials**-
One of the major problems in combinatorial representation theory is providing a cancellation-free, positive formula for Kostka-Foulkes polynomials known also as Lusztig's q-analogue of weight multiplicities. Such a formula is known in type A due to the remarkable Lascoux-Schutzenberger charge statistic on semistandard Young tableaux from 1978. For other types, however, very little is known and a conjectural formula exists only in type C. In this talk I discuss the classical result of Lascoux-Schutzenberger and I explain the conjectural formula of Lecouvey in type C. Our main result is a new combinatorial description of the statistic introduced by Lecouvey in the special case of the highest weight ω

_{1}. As an application we prove Lecouvey's conjectural formula in this case. This is a joint work with Thomas Gerber and Jacinta Torres. **Jacopo Gandini - Nilpotent Orbits of Height 2 and Involutions in the Affine Weyl Group**-
Let G be a semisimple algebraic group, with a fixed Borel subgroup B and maximal torus T in B. To any set of pairwise strongly orthogonal roots I will attach a nilpotent B-orbit in the Lie algebra of G, and will explain how the combinatorics of the involutions in the affine Weyl group of G relates to the geometry of such B-orbits. The talk is based on joint work with P. Moseneder Frajria and P. Papi.

**Marco D'Anna - Almost Gorenstein Rings and Further Generalizations: the One-Dimensional Case**-
I will first present the class of almost Gorenstein rings, focusing on the one-dimensional case.

This notion was introduced by Barucci and Froberg for algebroid curves and it has been recently generalized (by Goto and others) and widely studied for more general one-dimensional rings and, successively, in any dimension.

In the one-dimensional case, other possible generalizations of Gorenstein and almost Gorenstein has been proposed; these classes are close to Gorenstein and almost Gorenstein rings under different aspects.

I will describe one of this new classes, whose definition is motivated by the relationships between the properties of a local one-dimensional ring (R,m) and those of the R-algebra m:m. **Benjamin Drabkin - Containment-Tight Ideals from Singular Loci of Reflection Arrangements**-
Given an ideal I in a commutative Noetherian ring R, the m-th symbolic power of I is defined to be I

^{(m)}= ⋂_{p∈Ass(I)}(I_{p}^{m}∩ R). By results of Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede every ideal I of codimension e in a regular ring satisfies the containment I^{(er)}⊆ I^{r}. In many cases, this containment can be improved upon; however, in recent years a number of ideals have been found for which this containment is tight.

All known ideals exhibiting tight containments are codimension 2 and satisfy I^{(3)}⊄ I^{2}. Furthermore, most of these ideals define the singular loci of hyperplane arrangements for some complex reflection groups. This talk will aim to classify which complex reflection groups give rise to hyperplane arrangements whose singular loci exhibit the noncontainment I^{(3)}⊄ I^{2}. **Alexander Woo - Nash Blowups for Cominuscule Schubert Varieties**-
Nash blowups are a way of constructing birational maps that hopefully give you something less singular. We work out what this construction gives us for Schubert varieties on the Grassmannian and for other cominuscules. There is associated combinatorics from looking at the T-fixed points and T-fixed curves for Nash blowups and we work out what happens to those also. (This is joint work with Ed Richmond and William Slofstra and is https://arxiv.org/abs/1808.05918).

**Jaroslaw Buczynski - Constructions of K-Regular Maps Using Finite Local Schemes**-
A continuous map ℝ

^{m}→ ℝ^{N}or ℂ^{m}→ ℂ^{N}is called k-regular if the images of any k distinct points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, however there are very few results on the existence of such maps for particular values m. During the talk, using methods of algebraic geometry, we will construct k-regular maps. We will relate the upper bounds on the minimal value of N with the dimension of the a Hilbert scheme. The computation of the dimension of this space is explicit for k < 10, and we provide explicit examples for k at most 5. We will also provide upper bounds for arbitrary m and k. The problem has its interpretation in terms of interpolation theory: for a topological space X and a vector space V, a map X → V is k-regular if and only if the dual space V* embedded in space of continuous maps from X to the base field ℝ or ℂ is k-interpolating, i.e. for any k distinct points x_{1},…,x_{k}of X and any values f_{i}, there is a function in V*, which takes values f_{i}at x_{i}. Similarly, we can interpolate vector valued continuous functions, and analogous methods provide interesting results. **Federico Galetto - Representations with Finitely Many Orbits and Free Resolutions**-
The irreducible representations of reductive groups with finitely many orbits are parametrized by graded simple Lie algebras. For the exceptional types, W. Kraśkiewicz and J. Weyman computed minimal free resolutions for the coordinate rings of the orbit closures. I will describe a computational approach used to construct these resolutions and verify their exactness.

**Joachim Jelisiejew - Tensors, Commuting Matrices and Quot Schemes**-
Consider n-dimensional vector spaces A, B, C. Classification of tensors T in A ⊗ B ⊗ C that have minimal border rank (=are limits of tensors of the form ∑

_{i=1}^{n}a_{i}⊗ b_{i}⊗ c_{i}) is highly important in bounding the matrix multiplication exponent. This classification is naturally bonded to the geometry of Quot schemes and commuting matrices. In the talk I will explain the correspondence and some recent results on these schemes. No expertise with any of the mentioned objects is assumed. This is a joint work with Klemen Sivic.