We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.
The quantum Frobenius map and it splitting are shown to descend to corresponding maps for generalized $q$-Schur algebras at a root of unity. We also define analogs of $q$-Schur algebras for any affine algebra, and prove the corresponding results for them.
We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $FΣ_n$, which these connections reveal; graded categorification and connections with quantum groups and crystal bases; modular branching rules and the Mullineaux map; graded cellular structure and graded Specht modules; cuspidal systems for affine KLR algebras and imaginary Schur-Weyl duality, which connects representation theory of these algebras to the usual Schur algebras …
We study on Weyl modules of cyclotomic $q$-Schur algebras. In particular, we give the character formula of the Weyl modules by using the Kostka numbers and some numbers which are computed by a generalization of Littlewood-Richardson rule. We also study corresponding symmetric functions. Finally, we give some simple applications to modular representations of cyclotomic $q$-Schur algebras.
We obtain the structure of weight 2 blocks and [2:1]-pairs of q-Schur algebras, and compute explicitly the modular Alvis-Curtis duality for weight 2 blocks of finite general linear groups in non-defining characteristic.
We introduce Brauer characters for representations of the bismash products of groups in characteristic p > 0, p not 2 and study their properties analogous to the classical case of finite groups. We then use our results to extend to bismash products a theorem of Thompson on lifting Frobenius-Schur indicators from characteristic p to characteristic 0
The reduced Schur functions are studied. Their relations to the basic representation of $A^(1)_{r-1}$ and modular representations of the symmetric groups are clarified. Littlewood-Richardson coefficients appear in the linear relations among reduced Schur functions.
We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z(C) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholz…
In this note we give an account of recent progress on the construction of holomorphic vertex algebras as cyclic orbifolds as well as related topics in lattices and modular categories. We present a novel computation of the Schur indicator of a lattice involution orbifold using finite Heisenberg groups and discriminant forms.
In this paper we study the integral form of the lattice vertex algebra $V_L$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant $\mathbb Z$-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras.
Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories generated by a single non-invertible simple object: near groups, those fusion categories with one non-invertible simple object, and Haagerup-Izumi categories, those with one non-invertible simple object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariant…
We introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. As an application we obtain the explicit character formulas of simple affine vertex algebras associated with the Deligne exceptional series at level $-h^{\vee}/6-1$, which express the homogeneous Schur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Ree…
In this paper, we obtain a canonical central element $ν_H$ for each semi-simple quasi-Hopf algebra $H$ over any field $k$ and prove that $ν_H$ is invariant under gauge transformations. We show that if $k$ is algebraically closed of characteristic zero then for any irreducible representation of $H$ which affords the character $χ$, $χ(ν_H)$ takes only the values 0, 1 or -1, moreover if $H$ is a Hopf algebra or a twisted quantum double of a finite group then $χ(ν_H)$ is the corresponding Frobenius-Schur Indicator. We also prove an analog of a Theorem of Larson-Radford for split semi-simple quasi-…
We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine $\mathfrak{sl}_n$ and $\mathfrak{gl}_n$. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation…
We show that every central simple algebra A over a field k is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field (that is- A is Hopf Schur). If the characteristic of the field is zero, or if the algebra has a Galois splitting field of degree prime to the characteristic of k, we can take this Hopf algebra to be semisimple. We also show that if F is any finite extension of k, then F is a quotient of a finite dimensional Hopf algebra over k. We use it in order to show why the algebric closeness assumption is necessary in a weak form of Kaplansky's tenth conje…
We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group $\mathbf{U}_{\mathfrak{q}\tilde{\mathfrak{q}}}(\mathfrak{g}_\mathbb{R})$ of the modular double …
We study algebraic aspects of Kontsevich integrals as generating functions for intersection theory over moduli space and review the derivation of Virasoro and KdV constraints. 1. Intersection numbers 2. The Kontsevich integral 2.1. The main theorem 2.2 Expansion of Z on characters and Schur functions 2.3 Proof of the first part of the Theorem 3. From Grassmannians to KdV 4. Matrix Airy equation and Virasoro highest weight conditions 5. Genus expansion 6. Singular behaviour and Painlev'e equation. 7. Generalization to higher degree potentials
We give a modular branching rule for certain wreath products as a generalization of Kleshchev's modular branching rule for the symmetric groups. Our result contains a modular branching rule for the complex reflection groups $G(m,1,n)$ (which are often called the generalized symmetric groups) in splitting fields for $\mathbb{Z}/m\mathbb{Z}$. Especially for $m=2$ (which is the case of the Weyl groups of type $B$), we can give a modular branching rule in any field. Our proof is elementary in that it is essentially a combination of Frobenius reciprocity, Mackey theorem, Clifford's theory and Klesh…
We construct the positive principal series representations for $U_q(g_R)$ where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where $r$ is the rank of $g$. We show that under the representations, the generators of the Langlands dual group $U_{\tilde{q}}({}^Lg_R)$ are related to the generators of $U_q(g_R)$ by the transcendental relations. We define the modified quantum group of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the $q$-tori polynomials.
We present several infinite families of potential modular data motivated by examples of Drinfeld centers of quadratic categories. In each case, the input is a pair of involutive metric groups with Gauss sums differing by a sign, along with some conditions on the fixed points of the involutions and the relative sizes of the groups. From this input we construct $S$ and $T$ matrices which satisfy the modular relations and whose Verlinde coefficients are non-negative integers. We also check certain restrictions coming from Frobenius-Schur indicators. These families generalize Evans and Gannon's co…
We prove that the kernel of the natural action of the modular group on the center of the Drinfel'd double of a semisimple Hopf algebra is a congruence subgroup. To do this, we introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.
We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhauser, and Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius-Schur (FS-)exponent of a pivo…
A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We compute this formulas and discuss reductions and refinements for modular categories related with SU(N).Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer.We introduce the notion of a sp…
In this paper, we study modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of $SL_2(\mathbb{Z})$ associated with transitive modular categories are proven to be minimal and irreducible. Together with the Verlinde formula, we characterize prime transitive modular categories as the Galois conjugates of the adjoint subcategory of the quantum group modular category $\mathcal{C}(\mathfrak{sl}_2,p-2)$ for some prime $p > 3$. As a consequence, we complet…
