Schur Algebras Research Articles

A diagrammatic categorification of the affineq-Schur algebra S(n,n) forn≥ 3

This is a follow-up to the paper in which we categorified the affine quantum Schur algebra S(n,r) for 2 < r < n, using a quotient of Khovanov and Lauda's categorification of the affine quantum sl_n. In this paper we categorify S(n,n) for n > 2, using an extension of the aforementioned quotient.

Quantum affine [formula omitted] via Hecke algebras

The quantum loop algebra of gln is the affine analogue of quantum gln. In the seminal work [1], Beilinson–Lusztig–MacPherson gave a beautiful realisation for quantum gln via a geometric setting of quantum Schur algebras. More precisely, they used quantum Schur algebras to construct a certain algebra U in [1, 5.4] and proved in [1, 5.7] that U is isomorphic to quantum gln. We will present in this paper a full generalisation of BLM's realisation to the affine case. Though the realisation of the quantum loop algebra of gln is motivated by the work [1] for quantum gln, our approach is purely algebraic and combinatorial, independent of the geometric method for quantum gln. As an application, we discover a presentation of the Ringel–Hall algebra of a cyclic quiver by semisimple generators and their multiplications by the defining basis elements.

Quiver Schur algebras for linear quivers

We define a graded quasi-hereditary covering of the cyclotomic quiver Hecke algebras R n Λ of type A when e = 0 (the linear quiver) or e > n . We prove that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When e = 0 , we show that the Khovanov–Lauda–Rouquier grading on the quiver Hecke algebras is compatible with the Koszul grading on the blocks of parabolic category O Λ given by Backelin, building on the work of Beilinson, Ginzburg and Soergel. As a consequence, e = 0 our cyclotomic quiver Schur algebras are Koszul over fields of characteristic zero. Finally, we give an Lascoux–Leclerc–Thibon-like algorithm for computing the graded decomposition numbers of the cyclotomic quiver Schur algebras in characteristic zero.

Decomposition numbers of quantized walled Brauer algebras

In this paper, we establish an explicit relationship between decomposition numbers of quantized walled Brauer algebras and those for either Hecke algebras associated to certain symmetric groups or (rational) \(q\)-Schur algebras over a field \(\kappa \). This enables us to use Ariki’s result (J Math Kyoto Univ 36(4):789–808, 1996) and Varagnolo–Vasserot’s result (Duke Math J 100(2):267–297, 1999) to compute such decomposition numbers via inverse Kazhdan–Lusztig polynomials associated with extended affine Weyl groups of type \(A\) if the ground field is \({\mathbb {C}}\).

Polynomial representations of $$\mathrm{GL }(n)$$ GL ( n ) and Schur–Weyl duality

Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur–Weyl duality is the key for an equivalence between both categories.

Sheaves on Nilpotent Cones, Fourier Transform, and a Geometric Ringel Duality

The ”Ringel dual” of a quasi-hereditary algebra A is a new quasi-hereditary algebra A′ that is derived-equivalent to A, via a functor that sends tilting A-modules to projective A′modules. Geometric versions of this notion—that is, derived equivalences of constructible sheaves that send tilting perverse sheaves to projective perverse sheaves—have been studied on flag varieties by Beilinson–Bezrukavnikov–Mirkovic, Yun, and others. In this talk, I will discuss geometric Ringel duality for the nilpotent cone in type A, using the Fourier-Sato transform. This construction leads to a new proof of Donkin’s algebraic result that the Schur algebra is self-Ringel-dual. (In types other than A, the same construction still gives a derived equivalence, but for now lacks a corresponding algebraic interpretation.) This is joint work with C. Mautner.

Blocks of affine quantum Schur algebras

The affine quantum Schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine gln. Finite dimensional irreducible modules for the affine quantum Schur algebra S▵(n,r)v were classified in [15], where v∈C⁎ is not a root of unity. We will classify blocks of the affine quantum Schur algebra S▵(n,r)v in this paper.

Permanents, Doty coalgebras and dominant dimension of Schur algebras

A (hidden) multiplication on AZ(n,r), the Z-dual of the integral Schur algebra SZ(n,r) is explicitly constructed, possibly without a unit. The image of the multiplication map is shown to be spanned by bipermanents. Let k be any field of characteristic p>0. The image of the induced multiplication on Ak(n,r)=AZ(n,r)⊗Zk turns out to coincide with the Doty coalgebra Dn,r,p of truncated symmetric powers. Combined with a new straightening formula for bipermanents, it is proved that such a multiplication induces an isomorphism Ak(n,r)⊗Sk(n,r)Ak(n,r)≅Ak(n,r) as Sk(n,r)-bimodules if and only if r≤n(p−1), if and only if Dn,r,p=Ak(n,r). As a result, Sk(n,r) is a gendo-symmetric algebra, and its dominant dimension is at least two and admits a combinatorial characterization as long as r≤n(p−1).

INDUCTION AND RESTRICTION FUNCTORS FOR CYCLOTOMIC q-SCHUR ALGEBRAS

We define the induction and restriction functors for cyclotomic $q$-Schur algebras, and study some properties of them. As an application, we categorify a higher level Fock space by using the module categories of cyclotomic $q$-Schur algebras.

Quasi-hereditary algebras, exact Borel subalgebras, [formula omitted]-categories and boxes

Highest weight categories arising in Lie theory are known to be associated with finite dimensional quasi-hereditary algebras such as Schur algebras or blocks of category O. An analogue of the PBW theorem will be shown to hold for quasi-hereditary algebras: Up to Morita equivalence each such algebra has an exact Borel subalgebra. The category F(Δ) of modules with standard (Verma, Weyl, …) filtration, which is exact, but rarely abelian, will be shown to be equivalent to the category of representations of a directed box. This box is constructed as a quotient of a dg algebra associated with the A∞-structure on Ext⁎(Δ,Δ). Its underlying algebra is an exact Borel subalgebra.

Double coset algebras

For a finite monoid M with unit e and an indexed family G={Gi:i∈I} of subgroups of the group G(M) of invertible elements in M, the complex vector space A(M,G) with basis the double cosets of the form GimGj, m∈M, has a natural multiplication yielding an associative C-algebra which we call a double coset algebra. We construct two Z-algebras with identity, LGS(M,G) and RGS(M,G), called the left and right generalized Schur algebras, which as Z-modules are free with basis the double cosets. When M is the symmetric group Sr and G is the family of Young subgroups indexed by compositions of r with at most n parts, A(M,G) is isomorphic to the usual Schur algebra S(n,r) and LGS(M,G) corresponds to its standard Z-form.The structure constants we derive for these algebras provide multiplication rules useful for further analysis of the generalized Schur algebras. Our main result is then the observation that LGS(M,G) and RGS(M,G) are always Z-forms for A(M,G), that is, A(M,G)≅C⊗ZLGS(M,G)≅C⊗ZRGS(M,G).The Iwahori–Hecke algebra H(M,G) corresponding to a finite monoid M and subgroup G is isomorphic to the double coset algebra A(M,G) where G={G} consists of a single set. So, as a special case of our result, we obtain Z-forms LGS(M,G) and RGS(M,G) for arbitrary Iwahori–Hecke algebras. The existence of a Z-form for any such H(M,G) appears to be a new result.

On Schur algebras and related algebras VI: Some remarks on rational and classical Schur algebras

In a recent paper, Dipper and Doty, [4], introduced certain finite dimensional algebras, associated with the natural module of the general linear group and its dual, which they call rational Schur algebras. We give a proof, via tilting modules, that these algebras are in fact generalised Schur algebras. Using the same technique we show that certain finite dimensional algebras associated with classical groups, introduced by Doty, [20], are quasi-hereditary algebras. A generalised Schur algebras may be viewed as a quotient of the algebra of distributions of a reductive group by a certain ideal. We give generators for this ideal.

Homological properties of quantised Borel–Schur algebras and resolutions of quantised Weyl modules

We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The acyclicity of induction from some rank-one modules for quantised Borel–Schur subalgebras is deduced. This is used to prove the exactness of the complexes recently constructed by Boltje and Maisch, giving resolutions of the co-Specht modules for Hecke algebras.

Additive versus abelian 2-Representations of Fiat 2-Categories

We study connections between additive and abelian 2-rep- resentations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2- representation. We also describe, in detail, examples of fiat 2-categories associated to sl2-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.

The length function and matrix algebras

By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field we mean the least positive integer k such that words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, we study the main ring-theoretical properties of the length function: the behavior of the length under unity adjunction, direct sum of algebras, passage to subalgebras and homomorphic images. We give an upper bound for the length of the algebra as a function of the nilpotency index of its Jacobson radical and the length of the quotient algebra. We also provide examples of length computation for certain algebras, in particular, for the following classical matrix subalgebras: the algebra of upper triangular matrices, the algebra of diagonal matrices, the Schur algebra, Courter’s algebra, and for the classes of local and commutative algebras.

Sign sequences and decomposition numbers

We obtain a closed formula for the v-decomposition numbers dλμ(v) arising from the canonical basis of the Fock space representation of Uv(ŝle), where the partition λ is obtained from μ by moving some nodes in its Young diagram, all of which having the same e-residue. We also show that when these v-decomposition numbers are evaluated at v = 1, we obtain the corresponding decomposition numbers for the Schur algebras and symmetric groups.

Integral affine Schur–Weyl reciprocity

Let D▵(n) be the double Ringel–Hall algebra of the cyclic quiver △(n) and let D▵̇(n) be the modified quantum affine algebra of D▵(n). We will construct an integral form D▵̇(n)Z for D▵̇(n) such that the natural algebra homomorphism from D▵̇(n)Z to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct the integral form UZ(gl̂n) of the universal enveloping algebra U(gl̂n) of the loop algebra gl̂n=gln(Q)⊗Q[t,t−1], and prove that the natural algebra homomorphism from UZ(gl̂n) to the affine Schur algebra over Z is surjective. In a subsequent paper (Fu [10]), we will use affine Schur algebras to give BLM realization of UZ(gl̂n), and this enables us to give a new proof of the statements about UZ(gl̂n) given in this paper.

A class of perverse sheaves on framed representation varieties of the Jordan quiver

We study a class of simple perverse sheaves on framed representation varieties of the Jordan quiver in analogy with the one in Li (2013) [17]. We show that the associated local systems are always trivial. We use this class and the top Borel–Moore homology of certain related Steinberg-type varieties to give a geometric realization of the group algebra of a product of symmetric groups, a tensor product of Schur algebras, and a tensor product of Fock spaces.

Presenting Schur superalgebras

We provide a presentation of the Schur superalgebra and its quantum analogue which generalizes the work of Doty and Giaquinto for Schur algebras. Our results include a basis for these algebras and a presentation using weight idempotents in the spirit of Lusztig's modified quantum groups.

Endomorphism Algebras of Some Modules for Schur Algebras and Representation Dimension

We consider the representation dimension, for fixed n ≥ 2, of ordinary and quantised Schur algebras S(n, r) over a field k. For k of positive characteristic p we give a lower bound valid for all p. We also give an upper bound in the quantum case, when k has characteristic 0.