Harshit Yadav

We develop pivotal and spherical versions of graded extension theory. We define the corresponding analogues of Brauer-Picard 2-categorical groups and realize them as fixed points of natural \(\mathbb{Z}\) and \(\mathbb{Z}/2\mathbb{Z}\) 2-categorical actions. We classify graded extensions of a pivotal tensor category by monoidal 2-functors into the pivotal Brauer-Picard 2-categorical group. A similar statement is proven for spherical (unimodular) tensor categories. We also develop an obstruction theory for determining when pivotal and spherical structures can be extended.

We call a tensor functor \(F:\mathcal{C}\to\mathcal{D}\) between finite tensor categories \(\otimes\)-Frobenius if its left and right adjoints are isomorphic as \(\mathcal{C}\)-bimodule functors; this holds if and only if the centralizer \(Z({}_F\mathcal{D}_F)\) is unimodular. For perfect functors, pulling back a \(\mathcal{D}\)-module category along \(F\) preserves exactness, and pivotality, unimodularity, and sphericality are preserved when \(F\) is \(\otimes\)-Frobenius. Applications include criteria for \(\otimes\)-Frobenius functors arising from Hopf algebra maps, conditions for internal natural transformations to yield Frobenius algebras in \(\mathcal{Z}(\mathcal{C})\), and proofs that central tensor functors are Frobenius iff \(\otimes\)-Frobenius, and that any tensor functor between separable fusion categories is \(\otimes\)-Frobenius.

Inspired by the study of vertex operator algebra extensions, we answer the question of when the category of local modules over a commutative exact algebra in a braided finite tensor category is a (non-semisimple) modular tensor category. Along the way we provide sufficient conditions for the category of local modules to be rigid, pivotal and ribbon. We also discuss two ways to construct such commutative exact algebras. The first is the class of simple current algebras and the second is using right adjoints of central tensor functors. Furthermore, we discuss Witt equivalence and its relation with extensions of VOAs.

Due to the work of Shimizu (2019), various nondegeneracy conditions for braided finite tensor categories are equivalent. This theory is partially extended to braided module categories here. We introduce when a braided module category is "nondegenerate" and "factorizable", and establish that these properties are equivalent. The proof involves a new monadicity result for module categories. Lastly, we examine the Hopf case, using Kolb's (2020) notion of a quasitriangular comodule algebra to introduce "factorizable" comodule algebras. We then show that the representation category of a quasitriangular comodule algebra is nondegenerate in our sense precisely when the comodule algebra is factorizable. Several examples are provided.

Let \(A\) be a commutative algebra in a braided monoidal category \(\mathcal{C}\); e.g., \(A\) could be an extension of a vertex operator algebra (VOA) \(V\) in a category \(\mathcal{C}\) of \(V\)-modules. We study when the category \(\mathcal{C}_A\) of \(A\)-modules in \(\mathcal{C}\) and its subcategory \(\mathcal{C}_A^{\mathrm{loc}}\) of local modules inherit rigidity from \(\mathcal{C}\), and find conditions for \(\mathcal{C}\) and \(\mathcal{C}_A\) to inherit rigidity from \(\mathcal{C}_A^{\mathrm{loc}}\). First, we assume \(\mathcal{C}\) is a braided finite tensor category and prove rigidity of \(\mathcal{C}_A\) and \(\mathcal{C}_A^{\mathrm{loc}}\) under conditions based on criteria of Etingof-Ostrik for \(A\) to be an exact algebra. As a corollary, if \(A\) is a simple \(\mathbb{Z}_{\geq 0}\)-graded VOA with a strongly rational vertex operator subalgebra \(V\), then \(A\) is strongly rational, without requiring the categorical dimension of \(A\) as a \(V\)-module to be nonzero. Next, we assume \(\mathcal{C}\) is a Grothendieck-Verdier category. We first prove \(\mathcal{C}_A\) is also a Grothendieck-Verdier category. Using this, we prove that if \(\mathcal{C}_A^{\mathrm{loc}}\) is rigid, then so is \(\mathcal{C}\), under mild nondegeneracy and locality assumptions.

Let \(\mathcal{C}\) be a finite tensor category and \(\mathcal{M}\) an exact left \(\mathcal{C}\)-module category. We call \(\mathcal{M}\) unimodular if the finite multitensor category \(\mathrm{Rex}_{\mathcal{C}}(\mathcal{M})\) of right exact \(\mathcal{C}\)-module endofunctors of \(\mathcal{M}\) is unimodular. We provide various characterizations, properties, and examples of unimodular module categories. As our first application, we employ unimodular module categories to construct (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category. Our second application is a classification of unimodular module categories over the category of representations of a finite-dimensional Hopf algebra, answering a question of Shimizu. Using this, we provide an example of a finite tensor category whose categorical Morita equivalence class does not contain any unimodular tensor category.

Let \(U:\mathcal{C}\to\mathcal{D}\) be a strong monoidal functor between abelian monoidal categories admitting a right adjoint \(R\), such that \(R\) is exact, faithful and the adjunction \(U\dashv R\) is coHopf. Building on the work of Balan, we show that \(R\) is separable (resp., special) Frobenius monoidal if and only if \(R(\mathbf{1}_{\mathcal{D}})\) is a separable (resp., special) Frobenius algebra in \(\mathcal{C}\). If further, \(\mathcal{C}\), \(\mathcal{D}\) are pivotal (resp., ribbon) categories and \(U\) is a pivotal (resp., braided pivotal) functor, then \(R\) is a pivotal (resp., ribbon) functor if and only if \(R(\mathbf{1}_{\mathcal{D}})\) is a symmetric Frobenius algebra in \(\mathcal{C}\). As an application, we construct Frobenius monoidal functors into the Drinfeld center \(\mathcal{Z}(\mathcal{C})\), producing Frobenius algebras in it.

A Frobenius algebra is a finite-dimensional algebra \(A\) which comes equipped with a coassociative, counital comultiplication map \(\Delta\) that is an \(A\)-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius) algebras. We show that large classes of such algebras, including finite-dimensional weak Hopf algebras, come equipped with a nonzero map \(\Delta\) as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.

We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: a filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of a categorical Frobenius form. These two results of independent interest are then used to achieve our goal. As an application, we show that any exact module category over a symmetric finite tensor category \(\mathcal{C}\) is represented by a Frobenius algebra in \(\mathcal{C}\).

This chapter concerns edge labeled Young tableaux, introduced by H. Thomas and the third author. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences from combinatorics, algebraic and symplectic geometry, linear algebra, and computational complexity. We report on a new shifted analogue of edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule for the structure constants of the D. Anderson–W. Fulton ring, which is related to the equivariant cohomology of isotropic Grassmannians.

We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory.