VT Algebra Seminar — Spring 2022

A degree d curve neighborhood of a subvariety V in a smooth variety X is the closure of the degree d curves that intersect V. Curve neighborhoods where introduced by Buch, Chaput, Mihalcea, and Perrin to study quantum K-theory. The odd symplectic Grassmannian IG is a quasi-homogeneous space with homogeneous-like behavior. A very limited description of curve neighborhoods of Schubert varieties in IG was used by Mihalcea and myself to prove an (equivariant) quantum Chevalley rule. I will discuss how to give a full description of the irreducible components of curve neighborhoods of Schubert varieties in IG. One possible application is to use curve neighborhoods to calculate the minimum degree that occurs in a quantum product. This is joint work with Clelia Pech.

We will show that every finite abelian group arises as the group of rational points of an ordinary abelian variety over a finite field with 2, 3 or 5 elements. Similar results hold over finite fields of larger cardinality. On our way to proving these results, we will view the group of rational points of an abelian variety as a module over its endomorphism ring. By describing this module structure in important cases, we obtain (a fortiori) an understanding of the underlying groups. Combining this description of structure with recent results on the cardinalities of groups of rational points of abelian varieties over finite fields, we will deduce the main theorem. This work is joint with Stefano Marseglia.

We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard $q$-deformed enveloping algebra $U_q'(\mathfrak{so}_n)$ defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra $U(\mathfrak{so}_n)$. In this paper we only consider the case when $q$ is not a root of unity, and $q\to 1$ for the classical case. Extending work by Mazorchuk on $\mathfrak{so}_n$, we provide rational matrix coefficients for these infinite-dimensional modules of both $U_q'(\mathfrak{so}_n)$ and $U(\mathfrak{so}_n)$. We use these modules with rationalized formulas to embed the respective algebras into skew group algebras of shift operators. Casimir elements of $U_q'(\mathfrak{so}_n)$ were given by Gavrilik and Iorgov, and we consider the commutative subalgebra $\Gamma \subset U_q'(\mathfrak{so}_n)$ generated by these elements and the corresponding subalgebra $\Gamma_1 \subset U(\mathfrak{so}_n)$. The images of $\Gamma$ and $\Gamma_1$ under their respective embeddings into skew group algebras are equal to invariant algebras under certain group actions. We use these facts to show $\Gamma$ is a Harish-Chandra subalgebra of $U_q'(\mathfrak{so}_n)$ and $\Gamma_1$ is a Harish-Chandra subalgebra of $U(\mathfrak{so}_n)$.

In the 1980s, Greene introduced a finite field version of hypergeometric functions, now called Gaussian hypergeometric functions. Alternate versions of these functions were developed by Katz and McCarthy. In this talk, we will develop Greene’s functions in detail, comparing to classical hypergeometric functions along the way. We will highlight ways they relate to point-counting, traces of Hecke operators, and supercongruences. We then will present a systematic approach for translating some classical hypergeometric identities and evaluations to the finite field setting by an explicit dictionary.

TBA

Abstract: This talk is about comparing two a priori different quantizations of the GL_n-character variety for the torus. One of them is the spherical double affine Hecke algebra at q=t. The other is the combinatorial quantization defined by Alekseev, Grosse, and Schomerus, which has also reappeared in works of Varagnolo-Vasserot and Jordan. Conjecturally, both algebras are isomorphic. However, neither algebra possesses a presentation via generators and relations, and I will present a way to compare the two via an action on characters for GL_n. This approach has a natural t-deformation wherein the characters are replaced with Macdonald polynomials.