On $\tau$-tilting Finiteness of Symmetric Algebras of Polynomial Growth

We study the relationship between the cohomology of $q$-Schur algebras and Hecke algebras in type $A$. A key tool, which has independent interest, involves a specific Hecke algebra resolution, introduced in a special case by Deodhar. This resolution is somewhat akin to the Koszul resolution for Lie algebra cohomology, in that it has an explicit description yet applies for all modules. An application of the properties of this resolution establishes a ?quantum Weyl reciprocity? connecting, in a range of degrees, the cohomology of the $q$-Schur algebra with that of the Hecke algebra. This work also provides, independently of the resolution, cohomological inequalities which apply in all degrees and which sometimes can be related to Kazhdan?Lusztig polynomials. Finally, we extend to cohomology a decomposition number method of Erdmann, and illustrate its usefulness in computations. This method enables us to exhibit some properties of symmetric group cohomology which, if true, would imply the $SL_n$-version of Lusztig's characteristic $p$ character conjecture.

Let G G be a finite group of Lie type. In studying the nondefining characteristic (or cross-characteristic) representation theory of G G , the (specialized) Hecke algebra H = End G ⁡ ( Ind B G ⁡ 1 B ) H=\operatorname {End}_G(\operatorname {Ind}_B^G1_B) has played an important role. In particular, when G = G L n ( F q ) G=GL_n(\mathbb F_q) is a finite general linear group, this approach led to the Dipper-James theory of q q -Schur algebras A A . These algebras can be constructed over Z ≔ Z [ t , t − 1 ] \mathcal {Z}≔\mathbb Z[t,t^{-1}] as the q q -analog (with q = t 2 q=t^2 ) of an endomorphism algebra larger than H H , involving parabolic subgroups. The algebra A A is quasi-hereditary over Z \mathcal {Z} . An analogous algebra, still denoted A A , can always be constructed in other types. However, these algebras have so far been less useful than in the G L n GL_n case, in part because they are not generally quasi-hereditary. Several years ago, reformulating a 1998 conjecture, the authors proposed (for all types) the existence of a Z \mathcal {Z} -algebra A + A^+ having a stratified derived module category, with strata constructed via Kazhdan-Lusztig cell theory. The algebra A A is recovered as A = e A + e A=eA^+e for an idempotent e ∈ A + e\in A^+ . A main goal of this monograph is to prove this conjecture completely. The proof involves several new homological techniques using exact categories. Following the proof, we show that A + A^+ does become quasi-hereditary after the inversion of the bad primes. Some first applications of the result—e.g., to decomposition matrices—are presented, together with several open problems.

We prove that all wild blocks of type Hecke algebras with quantum characteristic — that is, blocks of weight at least 2 — are strictly wild, with the possible exception of the weight 2 Rouquier block for . As a corollary, we show that for , all wild blocks of the ‐Schur algebras are strictly wild, without exception.

On Schur algebras and related algebras V: some quasi-hereditary algebras of finite type

We formulate a $q$-Schur algebra associated with an arbitrary $W$-invariant finite set $X_{\text{f}}$ of integral weights for a complex simple Lie algebra with Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schur algebra and Hecke algebra associated with $W$. We then realize geometrically the $q$-Schur algebra and duality and construct a canonical basis for the $q$-Schur algebra with positivity. With suitable choices of $X_{\text{f}}$ in classical types, we recover the $q$-Schur algebras in the literature. Our $q$-Schur algebras are closely related to the category ${\mathcal{O}}$, where the type $G_{2}$ is studied in detail.

Many connections and dualities in representation theory and Lie theory can be explained using quasi-hereditary covers in the sense of Rouquier. Recent work by the first-named author shows that relative dominant (and codominant) dimensions are natural tools to classify and distinguish distinct quasi-hereditary covers of a finite-dimensional algebra. In this paper, we prove that the relative dominant dimension of a quasi-hereditary algebra, possessing a simple preserving duality, with respect to a direct summand of the characteristic tilting module is always an even number or infinite and that this homological invariant controls the quality of quasi-hereditary covers that possess a simple preserving duality. To resolve the Temperley–Lieb algebras, we apply this result to the class of Schur algebras $S(2, d)$ and their $q$ -analogues. Our second main result completely determines the relative dominant dimension of $S(2, d)$ with respect to $Q=V^{\otimes d}$ , the $d$ -th tensor power of the natural two-dimensional module. As a byproduct, we deduce that Ringel duals of $q$ -Schur algebras $S(2,d)$ give rise to quasi-hereditary covers of Temperley–Lieb algebras. Further, we obtain precisely when the Temperley–Lieb algebra is Morita equivalent to the Ringel dual of the $q$ -Schur algebra $S(2, d)$ and precisely how far these two algebras are from being Morita equivalent, when they are not. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$ -Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley–Lieb algebra.

Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of g l n \mathfrak {gl}_n and its associated monomial basis, we investigate q q -Schur algebras S q ( n , r ) \mathbf {S}_q(n,r) as “little quantum groups". We give a presentation for S q ( n , r ) \mathbf {S}_q(n,r) and obtain a new basis for the integral q q -Schur algebra S q ( n , r ) S_q(n,r) , which consists of certain monomials in the original generators. Finally, when n ⩾ r n\geqslant r , we interpret the Hecke algebra part of the monomial basis for S q ( n , r ) S_q(n,r) in terms of Kazhdan-Lusztig basis elements.

We show that the theorem by Hemmer and Nakano, on uniqueness of Specht filtration multiplicities, can be proved working entirely with representations of symmetric groups, or Hecke algebras. Furthermore, we give a new proof that Schur algebras are quasi-hereditary provided the characteristic of the field is at least 5. Our tools are some more general results on stratifying systems.

Group algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and S_q(n,r) with n geqslant r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson–Lusztig–MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type AIII/AIV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated with any weight function. These bases are the counterparts of Lusztig’s bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson–Lusztig–MacPherson construction, which is first introduced by Fan–Li from a geometric viewpoint.

We extend the family of classical Schur algebras in typeAA, which determine the polynomial representation theory of general linear groups over an infinite field, to a larger family, the rational Schur algebras, which determine the rational representation theory of general linear groups over an infinite field. This makes it possible to study the rational representation theory of such general linear groups directly through finite dimensional algebras. We show that rational Schur algebras are quasihereditary over any field, and thus have finite global dimension.We obtain explicit cellular bases of a rational Schur algebra by a descent from a certain ordinary Schur algebra. We also obtain a description, by generators and relations, of the rational Schur algebras in characteristic zero.

We define and study an action of the symmetric group on the Yokonuma--Hecke algebra. This leads to the definition of two classes of algebras. The first one is connected with the image of the algebra of the braid group inside the Yokonuma--Hecke algebras, and in turn with an algebra defined by Aicardi and Juyumaya known as the algebra of braids and ties. The second one can be seen as new deformations of complex reflection groups of type G(d,p,n). We provide several presentations for both algebras and a complete study of their representation theories using Clifford theory.

A new approach is established to compare Hochschild cohomologies of an algebra and of its centralizer subalgebras. This approach is based on dominant dimension, a homological dimension that is shown to control the comparison in a precise sense for a large class of algebras including classical and quantized Schur algebras. For the same class of algebras, it is shown that derived equivalences preserve dominant dimension. This is applied to determine the dominant dimensions of q q -Schur algebras and of their blocks.

One breakthrough in the theory of quantum groups is the construction of the canonical bases for quantum groups by Lusztig and Kashiwara. For type A, there is a geometric construction for (idempotented) quantum group together with a canonical basis due to Beilinson, Lusztig and MacPherson (BLM) using a stabilization procedure on a family of quantum Schur algebras of type A. Two essential ingredients in their work are a multiplication formula and a monomial basis. In this dissertation, we provide a BLM-type construction for affine type C. We realize the affine q-Schur algebras of type C as an endomorphism algebra of a certain permutation module of affine Hecke algebras, and then establish a multiplication formula on the Schur algebra level. We provide a direct construction of monomial bases for Schur algebras, which is also adapted to produce monomial bases for affine type A. Via a BLM-type stabilization on the Schur algebras, we construct an algebra K^c admitting canonical basis. We obtain that (K^a, K^c) forms a quantum symmetric pair in the spirit of Letzter and Kolb, where Ka is a quantum group of affine type A. The affine type C construction above is associated to an involution on Dynkin diagrams of affine type A. For other three types of involutions, we construct similar stabilization algebras admitting compatible canonical bases.

Higher level affine Schur and Hecke algebras