VT Algebra Seminar — Spring 2024

Abstract: Let g be a semisimple Lie algebra and θ be an involution of g. The quantization (Uq(g), Uı) of the symmetric pair (g, gθ) was systematically developed by Letzter where Uq(g) is the Drinfeld-Jimbo quantum group and Uı is a coideal subalgebra of it. We ususally refer Uı as the ıquantum group. Over the last decade, many fundamental constructions in quantum groups have been generalized to ıquantum groups by Wang and his collaborators. In this talk, we will discuss the super analogue of Uı and introduce one specific family which unites ıquantum groups (non-super) of type AI and AII. We will also demonstrate an ıSchur duality between this specific family and the q-Brauer algebra. This duality can be viewed as a quantization of the classical duality between the orthosymplectic Lie superalgebra and the Brauer algebra. This is joint work with Weiqiang Wang.

In this talk, we discuss a finite-field analogue of the arithmetic-geometric mean sequence. Study of the classical version dates back to work of Lagrange and Gauss, and makes beautiful contact with approximations of pi, elliptic integrals, hypergeometric functions, and elliptic curves. In the study of AGM(F_q), directed graphs called “jellyfish swarms” naturally arise. In studying these graphs, we find connections to both finite-field hypergeometric functions and elliptic curves over finite fields. Such connections give rise to new identities for Gauss’ class numbers of positive definite binary quadratic forms and show that the sizes of “jellyfish” (connected components of these graphs) are in part dictated by the order of a prime above 2 in certain class groups.

The notion of rational smoothness in Schubert varieties has been of interest since the foundational paper of Kazhdan and Lusztig connecting singularities of Schubert varieties with representation theory. This talk is based on recent joint work with Brian Boe, where we identify the loci of rationally smooth points and of smooth points in Schubert varieties for the Kac-Moody group of type $\tilde{A}_2$. We hope the methods will provide insight applicable to other types, and in particular, to type $\tilde{A}_n$ for $n \geq 2$. Our work continues the study of the lookup conjecture initiated in earlier work of the authors, as well as more recent joint work with Wenjing Li on spiral Schubert varieties in type $\tilde{A}_2$.

Abstract: Optimizing non-convex functions over potentially non-convex is a difficult task, and often it can be difficult even to certify a feasible solution. Modern optimization solvers frequently are comfortable providing approximately feasible solutions to such problems. Due to operating in floating-point arithmetic, we are constrained to providing rational solutions. We demonstrate classes of cubic optimization problems where it is NP-Hard to determine if a rational feasible solution exists. We also show how nearly feasible solutions can have super objective values and other related results. Time permitting, we will discuss some recent work on the complexity of finding integer solutions in non-convex sets.

The notion of shellable simplicial complexes has proved extremely useful in algebraic combinatorics, commutative algebra, and combinatorial topology. As such, it has been much studied in the past four decades. It is a classical result that matroid complexes. that is, simplicial complexes formed by the class of independent subsets in a matroid, are shellable. This has some bearing on the study of linear block codes, especially in regard to their Betti numbers and generalized weight enumerator polynomials. We now know that q-matroids have close connections with rank metric codes in a manner similar to the connection between matroids and codes. A recent result establishes shellability of q-matroid complexes and also determines the homology of these complexes in many cases. The determination of homology has now been completed for arbitrary q-matroid complexes. We will outline these developments whlie making an attempt to keep the prerequisites at a minimum. The contents of this talk are based on a joint work with Rakhi Pratihar and Tovohery Randrianarisoa (2022) and also with Rakhi Pratihar, Tovohery Randrianarisoa, Hugues Verdure and Glen Wilson (2024).

TBA

Consider an even-dimensional complex vector space endowed with a symmetric nondegenerate form. The set of subspaces of maximal dimension forms a projective variety called the maximal orthogonal Grassmannian X=OG(n, 2n). Since X is a smooth projective variety, we will consider its K-theory group K(X). As a group, K(X) is a freely generated by the classes of Schubert subvarieties and the multiplication of such classes can be computed using known computational rules. The quantum K-theory ring, QK(X), is a deformation of K(X) by Z[[q]]. As a group, it is similarly freely generated by classes of Schubert subvarieties but a (potentially) different multiplicative structure. The multiplicative constants can be computed using the Gromov-Witten invariants in the Kontsevich Moduli space. In this talk, I will give a short survey of the constructions in this setup and discuss recent progress of computing Pieri formulas for Gromov-Witten invariants.

TBA

Abstract: Let H ⊆ B ⊆ G be Cartan and Borel subgroups in a connected reductive complex algebraic group, and let Z be the complement of the zero section of a line bundle on G/B corresponding to a dominant weight λ of H. Let χ be a cocharacter of H such that for every Weyl group element w ∈ W, the pairing wλ ⋅ χ is strictly positive. Let S = χ(C^×) and call X = S ∖ Z the weighted flag variety.

The torus T = H/S acts on X, which enables the study of T-equivariant cohomology of X. In the case where X = G/P, Graham proved that the equivariant structure constants with respect to a Schubert basis satisfy positivity with respect to a system of simple roots. In the case where G = C^× × GL_n and λ restricts to a fundamental weight from GL_n, Abe-Matsumura find the existence of a basis of H_T^*(X) and parameters in H_T^*(pt) satisfying a similar positivity. We generalize all positivity results to any G and λ, interpret the basis of H_T^*(X) as Poincaré dual to weighted Schubert varieties, and define the notion of weighted root to interpret geometrically the parameters in H_T^*(pt).

Triangular modular curves are a generalization of modular curves and arise from quotients of the complex upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves naturally parameterize abelian varieties, making them interesting arithmetic objects. In this talk we will introduce triangular modular curves via classical modular curves. We will also study the genus of these curves. This is joint work with John Voight.

In this talk, we will define K-theoretic invariants involving certain virtual Euler characteristics of sheaves over the quot scheme of a curve. We demonstrate that these invariants fit into a topological quantum field theory valued in Z[[q]]. Additionally, we will show that the genus-0 invariants recover the small quantum K-ring of Grassmannians, offering a new approach for finding explicit formulas. In particular, we use torus localisation to obtain a Vafa-Intriligator type formula for the virtual Euler characteristics over the quot schemes. This is based on a joint work with Ming Zhang.