The (Q, q)-Schur Algebra

In this paper we use the Hecke algebra of type B to define anew algebra S which is an analogue of the q-Schur algebra. Weshow that S has ‘generic’ basis which is independentof the choice of ring and the parameters q and Q. We then constructWeyl modules for S and obtain, as factor modules, a family ofirreducible S-modules defined over any field. 1991 MathematicsSubject Classification: 16G99, 20C20, 20G05.     

Specht Filtrations for Hecke Algebras of Type A

Let H_q(d) be the Iwahori–Hecke algebra of the symmetricgroup, where q is a primitive 1th root of unity. Using resultsfrom the cohomology of quantum groups and recent results aboutthe Schur functor and adjoint Schur functor, it is proved that,contrary to expectations, for l 4 the multiplicities in a Spechtor dual Specht module filtration of an H_q(d)-module are welldefined. A cohomological criterion is given for when an H_q(d)-modulehas such a filtration. Finally, these results are used to givea new construction of Young modules that is analogous to theDonkin–Ringel construction of tilting modules. As a corollary,certain decomposition numbers can be equated with extensionsbetween Specht modules. Setting q = 1, results are obtainedfor the symmetric group in characteristic p 5. These resultsare false in general for p = 2 or 3.     

A Matrix Setting for the q-Schur Algebra

The Schur algebra S(n, r) has a basis (described in [6, 2.3])consisting of certain elements _i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue S_q(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James [3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power T^r(M_n)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock[12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in [5].This corresponds to the parameter v in U(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(n, r), and we shall usually refer to it as the v-Schur algebra.All three algebras are associative and have multiplicative identities,and the base field will be the field of rational functions,Q(v), unless otherwise stated. The symbols n and r shall bereserved for the integers given in the definitions of thesethree algebras.     

Kazhdan-Lusztig Cells and the Murphy Basis

Let H be the Iwahori–Hecke algebra associated with S_n,the symmetric group on n symbols. This algebra has two importantbases: the Kazhdan–Lusztig basis and the Murphy basis.We establish a precise connection between the two bases, allowingus to give, for the first time, purely algebraic proofs fora number of fundamental properties of the Kazhdan–Lusztigbasis and Lusztig's results on the a-function. 2000 MathematicsSubject Classification 20C08.     

Modular Branching Rules and the Mullineux Map for Hecke Algebras of Type A

We prove the quantum version - for Hecke algebras H A_n of typeA at roots of unity - of Kleshchev's modular branching rulefor symmetric groups. This result describes the socle of therestriction of an irreducible H A_n-module to the subalgebraH A_n–1. As a consequence, we describe the quantum versionof the Mullineux involution describing the irreducible moduleobtained on twisting an irreducible module with the sign representation.1991 Mathematics Subject Classification: 20C05, 20G05.     

On the Hochschild cohomology of tame Hecke algebras

We explicitly calculate a projective bimodule resolution for a special biserial algebra giving rise to the Hecke algebra H_q(S₄){{\mathcal H}_q(S_4)} when q = −1. We then determine the dimensions of the Hochschild cohomology groups.     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

Characterization of the Mod 3 Cohomology of the Compact, Connected, Simple, Exceptional Lie Groups of Rank 6

It is shown that the mod 3 cohomology of a 1-connected, homotopyassociative mod 3 H-space that is rationally equivalent to theLie group E₆ is isomorphic to that of E₆ as an algebra. Moreover,it is shown that the mod 3 cohomology of a nilpotent, homotopy-associativemod 3 H-space that is rationally equivalent to E₆, and whosefundamental group localized at 3 is non-trivial, is isomorphicto that of the Lie group Ad E₆ as a Hopf algebra over the mod3 Steenrod algebra. It is also shown that the mod 3 cohomologyof the universal cover of such an H-space is isomorphic to thatof E₆ as a Hopf algebra over the mod 3 Steenrod algebra. 2000Mathematics Subject Classification 57T05, 57T10, 57T25.     

De Rham Cohomology and Hodge Decomposition For Quantum Groups

Let be one of the N²-dimensionalbicovariant first order differential calculi for the quantumgroups GL_q(N), SL_q(N), SO_q(N), or Sp_q(N), where q is a transcendentalcomplex number and z is a regular parameter. It is shown thatthe de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologiesof its left-coinvariant, its right-coinvariant and its (two-sided)coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) thecohomology ring is isomorphic to the coinvariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in thesecases. The main technical tool is the spectral decompositionof the quantum Laplace-Beltrami operator. 2000 MathematicalSubject Classification: 46L87, 58A12, 81R50.     

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W（2, 2） Algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.     

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.     

On the Hecke Equivariance of the Borcherds Isomorphism

The Borcherds isomorphism is proved to be Hecke equivariantif one considers multiplicative Hecke operators acting on theintegral weight meromorphic modular forms. This answers a partof a question of Borcherds (see ‘Automorphic forms onO_{s+2, 2}(R) and infinite products’, Invent. Math. 120 (1995)161–213, 17.10), using his suggestion to define the multiplicativeHecke operators. 2000 Mathematics Subject Classification 11F37.     

Hochschild (Co)Homology Dimension

In 1989 Happel asked the question whether, for a finite-dimensionalalgebra A over an algebraically closed field k, gl.dim A < if and only if hch.dim A < . Here, the Hochschild cohomologydimension of A is given by hch.dim A := inf{n N₀ | dim HHⁱ(A) = 0 for i > n}. Recently Buchweitz, Green, Madsen andSolberg gave a negative answer to Happel's question. They founda family of pathological algebras A_q for which gl.dim A_q = but hch.dim A_q = 2. These algebras are pathological in manyaspects. However, their Hochschild homology behaviors are notpathological any more; indeed one has hh.dim A_q = = gl.dimA_q. Here, the Hochschild homology dimension of A is given byhh.dim A := inf{n N₀ | dim HH_i(A) = 0 for i > n}. This suggestsposing a seemingly more reasonable conjecture by replacing theHochschild cohomology dimension in Happel's question with theHochschild homology dimension: gl.dim A < if and only ifhh.dim A < if and only if hh.dim A = 0. The conjecture holdsfor commutative algebras and monomial algebras. In the casewhere A is a truncated quiver algebra, these conditions areequivalent to the condition that the quiver of A has no orientedcycles. Moreover, an algorithm for computing the Hochschildhomology of any monomial algebra is provided. Thus the cyclichomology of any monomial algebra can be read off when the underlyingfield is characteristic 0.     

Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantumgeneralized Kac–Moody algebras. For a quantum generalizedKac–Moody algebra U_q(g), we first introduce the categoryO_int of U_q(g)-modules and prove its semisimplicity. Next, wedefine the notion of crystal bases for U_q(g)-modules in thecategory O_int and for the subalgebra . We then prove the tensor product rule and the existence theoremfor crystal bases. Finally, we construct the global bases forU_q(g)-modules in the category O_int and for the subalgebra . 2000 Mathematics Subject Classification17B37, 17B67.     

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

Homological properties of quantized coordinate rings of semisimple groups

We prove that the generic quantized coordinate ring _q(G) isAuslander-regular, Cohen–Macaulay, and catenary for everyconnected semisimple Lie group G. This answers questions raisedby Brown, Lenagan, and the first author. We also prove thatunder certain hypotheses concerning the existence of normalelements, a noetherian Hopf algebra is Auslander–Gorensteinand Cohen–Macaulay. This provides a new set of positivecases for a question of Brown and the first author.     

Row and Column Removal Theorems for Homomorphisms of Specht Modules and Weyl Modules

We prove a q-analogue of the row and column removal theorems for homomorphisms between Specht modules proved by Fayers and the first author [16]. These results can be considered as complements to James and Donkin’s row and column removal theorems for decomposition numbers of the symmetric and general linear groups. In this paper we consider homomorphisms between the Specht modules of the Hecke algebras of type A and between the Weyl modules of the q-Schur algebra.This research was supported by ARC grant DP0343023. The first author was also supported by a Sesqui Research Fellowship at the University of Sydney.     

Hochschild Cohomology and Representation-Finite Algebras

Using Grothendieck's semicontinuity theorem for half-exact functors,we derive two semicontinuity results on Hochschild cohomology.We apply these to show that the first Hochschild cohomogy groupof the mesh algebra of a translation quiver over a domain vanishesif and only if the translation quiver is simply connected. Wethen establish an exact sequence relating the first Hochschildcohomology group of an algebra to that of the endomorphism algebraof a module and apply it to study the first Hochschild cohomologygroup of an Auslander algebra. Our main result shows that fora finite-dimensional and representation-finite algebra algebraA over an algebraically closed field with Auslander algebra the following conditions are equivalent:

1. (1)A admits no outer derivation;
2. (2) admits no outer derivations;
3. (3) A is simply connected;
4. (4) is strongly simply connected.

. 2000 Mathematics Subject Classification 16E30, 16G30.     

Gromov-Witten Invariants of Flag Manifolds, VIA D-Modules

We present a method for computing the 3-point genus zero Gromov–Witteninvariants of the complex flag manifold G/B from the relationsof the small quantum cohomology algebra QH^*G/B (G is a complexsemisimple Lie group and B is a Borel subgroup). In [3] and[9], at least in the case G = GL_nC, two algebraic/combinatoricmethods have been proposed, based on suitably designed axioms.Our method is quite different, being differential geometricin nature; it is based on the approach to quantum cohomologydescribed in [7], which is in turn based on the integrable systemspoint of view of Dubrovin and Givental.     

Small Deviations of Riemann-Liouville Processes In lq Spaces With Respect To Fractal Measures

We investigate Riemann–Liouville processes R_H, with H> 0, and fractional Brownian motions B_H, for 0 < H <1, and study their small deviation properties in the spacesL_q([0, 1], µ). Of special interest here are thin (fractal)measures µ, that is, those that are singular with respectto the Lebesgue measure. We describe the behavior of small deviationprobabilities by numerical quantities of µ, called mixedentropy numbers, characterizing size and regularity of the underlyingmeasure. For the particularly interesting case of self-similarmeasures, the asymptotic behavior of the mixed entropy is evaluatedexplicitly. We also provide two-sided estimates for this quantityin the case of random measures generated by subordinators. While the upper asymptotic bound for the small deviation probabilityis proved by purely probabilistic methods, the lower bound isverified by analytic tools concerning entropy and Kolmogorovnumbers of Riemann–Liouville operators. 2000 MathematicsSubject Classification 60G15 (primary), 47B06, 47G10, 28A80(secondary).