Schur Algebras Research Articles

Tilting equivalences: From hereditary algebras to symmetric groups

We study Koszul duality for finite dimensional hereditary algebras, and various generalisations to trivial extension algebras, to Schur algebras, to doubles of Schur bialgebras, and to deformations of doubles of Schur bialgebras. We describe applications to the modular representation theory of symmetric groups.

The rational Schur algebra

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.

Schur functors on QF-3 standardly stratified algebras

Let A be a QF-3 standardly stratified algebra and f be a Schur functor corresponding to some projective-injective faithful A-module, denoted by Ae. The main result of this paper is to prove that, if the dominant dimension of A is sufficiently large, then f induces a full embedding from ℱ(Δ) to eAe-mod which preserves Ext-groups up to certain degrees, where ℱ(Δ) denotes the full subcategory of A-mod whose objects are filtered by standard A-modules. We check this criterion on some typical examples, quantized Schur algebras Sq(n, r) with n ≥ r and finite-dimensional algebras associated with the Bernstein-Gelfand-Gelfand category \( \mathcal{O} \) of semisimple complex Lie algebras.

Comparing GL n -representations by characteristic-free isomorphisms between generalized Schur algebras

Comparing GL n -representations by characteristic-free isomorphisms between generalized Schur algebras

On the blocks of semisimple algebraic groups and associated generalized Schur algebras

In this paper we give a new proof for the description of the blocks of any semisimple simply connected algebraic group when the characteristic of the field is greater than 5. The first proof was given by Donkin and works in arbitrary characteristic. Our new proof has two advantages. First we obtain a bound on the length of a minimum chain linking two weights in the same block. Second we obtain a sufficient condition on saturated subsets π of the set of dominant weights which ensures that the blocks of the associated generalized Schur algebra are simply the intersection of the blocks of the algebraic group with the set π. However, we show that this is not the case in general for the symplectic Schur algebras, disproving a conjecture of Renner.

Kleshchev's decomposition numbers and branching coefficients in the Fock space

We give combinatorial descriptions of some coefficients of the canonical basis of the q-deformed Fock space representation of Uq(sle) and of some matrix entries for the action of the Chevellay generators fr with respect to the canonical basis. These are q-analogues of results of Kleshchev on decomposition numbers and branching coefficients for symmetric groups and Schur algebras.

On Defining Ideals or Subrings of Hall Algebras

Let A be a finitary algebra over a finite field k, and A-\(\text{mod}\) the category of finite dimensional left A-modules. Let \(\mathcal{H}(A)\) be the corresponding Hall algebra, and for a positive integer r let D r (A) be the subspace of \(\mathcal{H}(A)\) which has a basis consisting of isomorphism classes of modules in A-\(\text{mod}\) with at least r + 1 indecomposable direct summands. If A is the path algebra of the quiver of type A n with linear orientation, then D r (A) is known to be the kernel of the map from the twisted Hall algebra to the quantized Schur algebra indexed by n + 1 and r. For any A, we determine necessary and sufficient conditions for D r (A) to be an ideal and some conditions for D r (A) to be a subring of \(\mathcal{H}(A)\). For A the path algebra of a quiver, we also determine necessary and sufficient conditions for D r (A) to be a subring of \(\mathcal{H}(A)\).

Every projective Schur algebra is Brauer equivalent to a radical abelian algebra

We prove that any projective Schur algebra over a field K is equivalent in Br(K) to a radical abelian algebra. This was conjectured in 1995 by Sonn and the first author of this paper. As a consequence we obtain a characterization of the projective Schur group by means of Galois cohomology. The conjecture was known for algebras over fields of positive characteristic. In characteristic zero the conjecture was known for algebras over fields with an Henselian valuation over a local or global field of characteristic zero.

Blocks of cyclotomic Hecke algebras

This paper classifies the blocks of the cyclotomic Hecke algebras of type G ( r , 1 , n ) over an arbitrary field. Rather than working with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic Schur algebras. We obtain an explicit description of the blocks by analyzing the combinatorics of ‘Jantzen equivalence.’ We remark that a proof of the classification of the blocks of the cyclotomic Hecke algebras was announced in 1999. Unfortunately, Cox has discovered that this previous proof is incomplete.

Group algebras of Kleinian type and groups of units

The algebras of Kleinian type are finite-dimensional semisimple rational algebras A such that the group of units of an order in A is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings RG, with R an order in a number field and G a finite group, such that the group of units of RG is virtually a direct product of free-by-free groups.

COHOMOLOGY GROUPS OF RADICAL EXTENSIONS

If k is a subfield of Q(em) then the cohomology group H2(k (en)/k) is isomorphic to H(k(en′ )/k) with gcd(m, n ′) = 1. This enables us to reduce a cyclotomic k-algebra over k(en) to the one over k(en′ ). A radical extension in projective Schur algebra theory is regarded as an analog of cyclotomic extension in Schur algebra theory. We will study a reduction of cohomology group of radical extension and show that a Galois cohomology group of a radical extension is isomorphic to that of a certain subextension of radical extension. We then draw a cohomological characterization of radical group.

On projective resolutions of simple modules over the Borel subalgebra [formula omitted] of the Schur algebra [formula omitted] for [formula omitted

We construct projective resolutions for simple modules over the Borel subalgebras S + ( 2 , r ) and S + ( 3 , r ) of the Schur algebras S ( 2 , r ) and S ( 3 , r ) over a field of positive characteristic. In the case S + ( 2 , r ) we get a minimal projective resolution. We find also projective resolutions of minimal possible length for Weyl modules over the Schur algebra S ( 2 , r ) , corresponding to the regular weights.

Extensions, Levi subgroups and character formulas

This paper consists of three interconnected parts. Parts I, III study the relationship between the cohomology of a reductive group G and that of a Levi subgroup H. For example, we provide a sufficient condition, arising from Kazhdan–Lusztig theory, for a natural map Ext G • ( L , L ′ ) → Ext H • ( L H , L H ′ ) to be surjective, given irreducible G-modules L , L ′ and corresponding irreducible H-modules L H , L H ′ . In cohomological degree n = 1 , the map is always an isomorphism, under our hypothesis. These results were inspired by recent work of Hemmer [D. Hemmer, A row removal theorem for the Ext 1 quiver of symmetric groups and Schur algebras, Proc. Amer. Math. Soc. 133 (2) (2005) 403–414 (electronic)] for G = GL n , and both extend and improve upon the latter when our condition is met. Part II obtains results on Lusztig character formulas (LCFs) for reductive groups, obtaining new necessary and sufficient conditions for such formulas to hold. In the special case of G = GL n , these conditions can be recast in a striking way completely in terms of explicit representation theoretic properties of the symmetric group. This work on GL n improves upon [B. Parshall, L. Scott, Quantum Weyl reciprocity for cohomology, Proc. London Math. Soc. 90 (3) (2005) 655–688], which established only sufficient, rather than necessary and sufficient, conditions for the validity of the LCF in terms of the symmetric group.

Hochschild cohomology of algebras with homological ideals

Let $\varphi: A \to B$ be a homological epimorphism of $k$-algebras. We investigate the relationship of the Hochschild cohomologies $H^{i}(A)$ and $H^{i}(B)$ of $A$ and $B$, and show that they can be connected by a long exact sequence. In particular, if $A$ is a quasihereditary algebra and $B$ is the quotient of $A$ by a minimal heredity ideal, then the long exact sequence provides information on $H^{i}(A)$, $H^{i}(B)$ and the extension groups between costandard modules and standard modules, thus one can actually compute $H^{i}(A)$ inductively. As a consequence, we obtain the Hochschild cohomology of all nonsemisimple Temperley-Lieb algebras and representation-finite Schur algebras.

Endomorphism rings of permutation modules over maximal Young subgroups

Let K be a field of characteristic two, and let λ be a two-part partition of some natural number r. Denote the permutation module corresponding to the (maximal) Young subgroup Σ λ in Σ r by M λ . We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra S K ( λ ) = 1 λ S K ( 2 , r ) 1 λ = End K Σ r ( M λ ) of the Schur algebra S K ( 2 , r ) . These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers.

On tensor spaces over Hecke algebras of type [formula omitted

Let W ( B n ) be the Weyl group of type B n and H ( B n ) be the associated Iwahori–Hecke algebra. In this paper, we study the n-tensor space V ⊗ n (where dim V = 2 m ) with natural actions (introduced in [R.M. Green, Hyperoctahedral Schur algebras, J. Algebra 192 (1997) 418–438]) of W ( B n ) and of H ( B n ) . For each composition λ = ( λ 1 , … , λ m ) of n, let e λ be the corresponding initial basis element of V ⊗ n (see (3.8) for definition). We show that, if d is a distinguished right coset representative of S λ in W ( B n ) , then the action of the natural basis element T d on e λ coincides with the * permutation action of d up to a scalar. As an application, we prove that the n-tensor space decomposes (at the integral level) into a direct sum of some permutation modules (over Hecke algebra H ( B n ) ) with respect to certain standard parabolic subalgebras.

Schur algebras of classical groups

Throughout this paper the base field will be C . By Doty's definition [S. Doty, Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998) 165–199], a Schur algebra of a classical group G is the image of the representation map C G → End C ( ( C n ) ⊗ r ) , where C n is the natural representation and r any natural number. These Schur algebras are semisimple over C . Firstly we determine when the Schur algebras are generalized Schur algebras in Donkin's sense (see [S. Donkin, On Schur algebras and related algebras, I, J. Algebra 104 (1986) 310–328]). The main step is to decompose the tensor space ( C n ) ⊗ r , using path model by Littelmann [P. Littelmann, A Littlewood–Richardson rule for symmetrizable Kac–Moody algebras, Invent. Math. 116 (1994) 329–346]. Secondly we relate Schur algebras with different parameters and form inverse systems from Schur algebras in the same type. We find the inverse limit naturally contains the universal enveloping algebra of the corresponding Lie algebra.

Schur Superalgebras in Characteristic p

The structure of a Schur superalgebra S = S(1 ∣ 1, r) in odd characteristic p is completely determined. The algebra S is semisimple if and only if p does not divide r. If p divides r, then simple S-modules are one-dimensional and the quiver and relations of S can be immediately seen from its regular representation computed in this paper. Surprisingly, if p divides r, then S is neither quasi-hereditary nor cellular nor stratified, as one would expect by analogy with classical Schur algebras or Schur superalgebras in characteristc zero.

Presenting generalized Schur algebras in types B, C, D

We give explicit presentations by generators and relations of certain generalized Schur algebras (associated with tensor powers of the natural representation) in types B, C, D. This extends previous results in type A obtained by two of the authors. The presentation is compatible with the Serre presentation of the corresponding universal enveloping algebra. In types C, D this gives a presentation of the corresponding classical Schur algebra (the image of the representation on a tensor power) since the classical Schur algebra coincides with the generalized Schur algebra in those types. This coincidence between the generalized and classical Schur algebra fails in type B, in general.

STRATIFYING SYSTEMS, FILTRATION MULTIPLICITIES AND SYMMETRIC GROUPS

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.