Quantum Schur algebras revisited

A geometric construction of quantum Schur algebras was given by Beilinson, Lusztig and MacPherson in terms of pairs of flags in a vector space. By viewing such pairs of flags as representations of a poset, we give a recursive formula for the structure constants of quantum Schur algebras which is related to certain Hall polynomials. As an application, we provide a direct proof of the fundamental multiplication formulas which play a key role in the Beilinson-Lusztig-MacPherson realization of quantum gl_n. In the appendix we show how to groupoidify quantum Schur algebras in the sense of Baez, Hoffnung and Walker.     

A note on bideterminants for Schur superalgebras

Let S(m|n,r)_Z be a Z-form of a Schur superalgebra S(m|n,r) generated by elements ξ_i,j. We solve a problem of Muir and describe a Z-form of a simple S(m|n,r)-module D_λ,Q over the field Q of rational numbers, under the action of S(m|n,r)_Z. This Z-form is the Z-span of modified bideterminants [T_?:T_i] defined in this work. We also prove that each [T_?:T_i] is a Z-linear combination of modified bideterminants corresponding to (m|n)-semistandard tableaux T_i.     

Simple finite-dimensional modular noncommutative Jordan superalgebras

We classify the central simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic $p>2$. The case of characteristic 0 was considered by the authors in the previous paper [21]. In particular, we describe Leibniz brackets on all finite dimensional central simple Jordan superalgebras except mixed (nor vector neither Poisson) Kantor doubles of the supercommutative superalgebra $B(m,n)$.     

BGP-reflection functors and cluster combinatorics

We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the “truncated simple reflections” on the set of almost positive roots Φ_≥−1 associated with a finite dimensional semi-simple Lie algebra. Combining this with the tilting theory in cluster categories developed in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054], we give a unified interpretation via quiver representations for the generalized associahedra associated with the root systems of all Dynkin types (simply laced or non-simply laced). This confirms the Conjecture 9.1 in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054] for all Dynkin types.     

Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x_1,1,…,x_n,n] to give a new construction of the Kazhdan-Lusztig representations of S_n. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x_1,1,…,x_n,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations.     

Maximal subalgebras of Jordan superalgebras

The maximal subalgebras of the finite-dimensional simple special Jordan superalgebras over an algebraically closed field of characteristic 0 are studied. This is a continuation of a previous paper by the same authors about maximal subalgebras of simple associative superalgebras, which is instrumental here.     

On herstein's constructions relating jordan and associative superalgebras

We investigate the Jordan structure of a prime associative superalgebra and the Jordan structure of the symmetric elements of a *-prime associative superalgebra with superinvolution.     

Tilting theory and cluster combinatorics

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.     

Generalized Jones traces and Kazhdan-Lusztig bases

We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the positivity of certain structure constants for the associated Kazhdan-Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading terms of certain Kazhdan-Lusztig polynomials.     

On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations

Using Du’s characterization of the dual canonical basis of the coordinate ring O(GL(n,C)), we express all elements of this basis in terms of immanants. We then give a new factorization of permutations w avoiding the patterns 3412 and 4231, which in turn yields a factorization of the corresponding Kazhdan-Lusztig basis elements of the Hecke algebra H_n(q). Using the immanant and factorization results, we show that for every totally nonnegative immanant and its expansion with respect to the basis of Kazhdan-Lusztig immanants, the coefficient d_w must be nonnegative when w avoids the patterns 3412 and 4231.     

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.     

Higher level affine Schur and Hecke algebras

We define a higher level version of the affine Hecke algebra and prove that, after completion, this algebra is isomorphic to a completion of Webster's tensor product algebra of type A. We then introduce a higher level version of the affine Schur algebra and establish, again after completion, an isomorphism with the quiver Schur algebra. An important observation is that the higher level affine Schur algebra surjects to the Dipper-James-Mathas cyclotomic q-Schur algebra. Moreover, we give nice diagrammatic presentations for all the algebras introduced in this paper.     

On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory

We describe the irreducible components of Springer fibers for hook and two-row nilpotent elements of as iterated bundles of flag manifolds and Grassmannians. We then relate the topology (in particular, the intersection homology Poincaré polynomials) of the pairwise intersections of these components with the inner products of the Kazhdan-Lusztig basis elements of irreducible representations of the rational Iwahori-Hecke algebra of type A corresponding to the hook and two-row Young shapes.     

Malcev superalgebras with trival nucleus

The nucleus of a Malcev superalgebra M measures how far it is from being a Lie superalgebraM being a Lie superalgebra if and only if its nucleus is the whole M. This paper is devoted to study Malcev superalgebras in the opposite direction, that is, with trivial nucleus. The odd part of any finite-dimensional Malcev superalgebra with trivial nucleus is shown to be contained in the solvable radical. For algebraically closed fields, any such superalgebra splits as the sum of its solvable radical and a semisimple Malcev algebra contained in the even part, which is a direct sum of copies of sl(2, F) and the seven-dimensional simple non-Lie Malcev algebra, obtained from the Cayley-Dickson algebra.     

Nonlinear Liouville theorems for some critical problems on H-type groups

In this paper, we provide a non-existence result for a semilinear sub-elliptic Dirichlet problem with critical growth on the half-spaces of any group of Heisenberg-type. Our result improves a recent theorem in (Math. Ann. 315 (3) (2000) 453).     

Lie superalgebras graded by the root system B(<Emphasis Type="Italic">m,n</Emphasis>)

We determine the Lie superalgebras that are graded by the root system B(m,n) of the orthosymplectic Lie superalgebra osp(2m + 1,2n). Mathematics Subject Classification (2000) Primary 17B70, Secondary 17A70     

Star-regularity and regular completions

In this paper we establish a new characterisation of star-regular categories, using a property of internal reflexive graphs, which is suggested by a recent result due to O. Ngaha Ngaha and the first author. We show that this property is, in a suitable sense, invariant under regular completion of a category in the sense of A. Carboni and E.M. Vitale. Restricting to pointed categories, where star-regularity becomes normality in the sense of the second author, this reveals an unusual behaviour of the exactness property of normality (i.e. the property that regular epimorphisms are normal epimorphisms) compared to other closely related exactness properties studied in categorical algebra.