discriminants of brauer algebras in this paper we compute gram determinants associated to all cell modules of brauer algebras fancyscript b n partial theoretically we know when a cell module of fancyscript b n partial is equal to its simple head this gives a solution of this long standing problem

minimal length elements in twisted conjugacy classes of finite coxeter groups let w be a finite coxeter group and let f be an automorphism of w that leaves the set of generators of w invariant we establish certain properties of elements of minimal length in the f conjugacy classes of w that allow us to define character tables for the corresponding twisted iwahori hecke algebras these results are extensions of results obtained by geck and pfeiffer in the case where f is trivial

smallest graded betti numbers it is known that given a hilbert function h there need not exist a module which has uniquely the smallest graded betti numbers among all modules attaining h in this paper we extend the previous example of this behavior to an infinite family and demonstrate with a second infinite family that even when the given hilbert function is that of a complete intersection a module with uniquely smallest graded betti numbers need not exist finally we prove a conjecture of geramita harima and shin concerning the non existence of uniquely smallest graded betti numbers among all gorenstein rings attaining a given hilbert function

a quantum analogue of the first fundamental theorem of classical invariant theory we establish a noncommutative analogue of the first fundamental theorem of classical invariant theory for each quantum group associated with a classical lie algebra we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action the subspace of invariants is shown to form a subalgebra which is finitely generated we determine generators of this subalgebra of invariants and determine their commutation relations in each case considered the noncommutative modules we construct are flat deformations of their classical commutative analogues our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory which follows from our results by taking the limit as q 1 our method similarly leads to a definition of quantum spheres which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry

separation of variables and the computation of fourier transforms on finite groups ii we present a general diagrammatic approach to the construction of efficient algorithms for computing the fourier transform of a function on a finite group by extending work which connects bratteli diagrams to the construction of fast fourier transform algorithms we make explicit use of the path algebra connection to the construction of gel fand tsetlin bases and work in the setting of quivers we relate this framework to the construction of a configuration space derived from a bratteli diagram in this setting the complexity of an algorithm for computing a fourier transform reduces to the calculation of the dimension of the associated configuration space our methods give improved upper bounds for computing the fourier transform for the general linear groups over finite fields the classical weyl groups and homogeneous spaces of finite groups while also recovering the best known algorithms for the symmetric group

irreducible modules for equivariant map superalgebras and their extensions abstract let be a group acting on a scheme x and on a lie superalgebra g both defined over an algebraically closed field of characteristic zero k the corresponding equivariant map superalgebra m g x is the lie superalgebra of equivariant regular maps from x to g in this paper we complete the classification of finite dimensional irreducible m g x modules when g is a finite dimensional simple lie superalgebra x is of finite type and is a finite abelian group acting freely on the rational points of x by classifying these m g x modules in the case where g is a periplectic lie superalgebra we also describe extensions between irreducible modules in terms of homomorphisms and extensions between modules for certain finite dimensional lie superalgebras as an application one obtains the block decomposition of the category of finite dimensional m g x modules in terms of blocks and spectral characters of finite dimensional lie superalgebras

irreducible finite dimensional representations of equivariant map algebras suppose a finite group acts on a scheme x and a finite dimensional lie algebra g the corresponding equivariant map algebra is the lie algebra m of equivariant regular maps from x to g we classify the irreducible finite dimensional representations of these algebras in particu lar we show that all such representations are tensor products of evaluation representations and one dimensional representations and we establish conditions ensuring that they are all evaluation representations for example this is always the case if m is perfect our results can be applied to multiloop algebras current algebras the onsager algebra and the tetrahedron algebra doing so we easily recover the known classifications of irreducible finite dimensional representations of these algebras moreover we obtain previously unknown classifica tions of irreducible finite dimensional representations of other types of equivariant map algebras such as the generalized onsager algebra

on tensor categories of lie type en n 9 let v be the minuscule representation of the lie algebra g en n 6 7 we show that the centralizer of the action of the quantum group uqg en on v n is generated by the r matrices and one additional element appearing in the n 1 st tensor power a similar description can be given for the classical limit an analogous statement is true for a certain direct summand of v n for uqg en with n 5 n 9 here v is the module whose highest weight is the fundamental weight labeled by the vertex furthest from the triple point of the graph en moreover we obtain a 2 parameter family of braid representations generalizing a q version of the brauer centralizer algebras

a ribbon hopf algebra approach to the irreducible representations of centralizer algebras the brauer birman wenzl and type a iwahori hecke algebras abstract we show how the ribbon hopf algebra structure on the drinfel d jimbo quantum groups of types a b c and d can be used to derive formulas giving explicit realizations of the irreducible representations of the iwahori hecke algebras of type a and the birman wenzl algebras we use this derivation to give explicit realizations of the irreducible representations of the brauer algebras as well the derivation is accomplished by way of a combination of techniques from operator algebras quantum groups and the theory of 3 manifold invariants although our applications are in the cases of the quantum groups of types a b c and d most of the aspects of our approach apply in the general setting of ribbon hopf algebras

combinatorics of transformations from standard to non standard bases in brauer algebras transformation coefficients between standard bases for irreducible representations of the brauer centralizer algebra and split bases adapted to the subalgebra f1 f2 f are considered after providing the suitable combinatorial background based on the definition of the i coupling relation on nodes of the subduction grid we introduce a generalized version of the subduction graph which extends the one given in chilla 2006 j phys a math gen 39 7657 for symmetric groups thus we can describe the structure of the subduction system arising from the linear method and give an outline of the form of the solution space an ordering relation on the grid is also given and then as in the case of symmetric groups the choices of the phases and of the free factors governing the multiplicity separations are discussed

an algorithm for characters of hecke algebras hn q of type an 1 recently the characters of irreducible representations of the hecke algebra hn q of type an 1 were identified with the transition coefficients relating q generalized power sum symmetric functions to schur functions king and wybourne 1990 using this result one obtains an easy combinatorial algorithm for calculating the characters

reverse lex ideals we study reverse lex ideals in a polynomial ring and compare their properties to those of lex ideals in particular we provide an analogue of green s theorem for reverse lex ideals we also compare the betti numbers of strongly stable and square free strongly stable monomial ideals to those of reverse lex ideals

a modified brauer algebra as centralizer algebra of the unitary group the centralizer algebra of the action of u n on the real tensor powers r r v of its natural module v c n is described by means of a modification in the multiplication of the signed brauer algebras the relationships of this algebra with the invariants for u n and with the decomposition of r r v into irreducible submodules is considered

derived equivalences and sl2 categorifications for uq gln abstract we give a construction of sl 2 categorifications for u q gl n for generic q and for q a root of unity

generalized characters of the symmetric group normalized irreducible characters of the symmetric group s n can be understood as zonal spherical functions of the gelfand pair s n s n diags n they form an orthogonal basis in the space of the functions on the group s n invariant with respect to conjugations by s n in this paper we consider a different gelfand pair connected with the symmetric group that is an unbalanced gelfand pair s n s n 1 diags n 1 zonal spherical functions of this gelfand pair form an orthogonal basis in a larger space of functions on s n namely in the space of functions invariant with respect to conjugations by s n 1 we refer to these zonal spherical functions as normalized generalized characters of s n the main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group the paper gives a murnaghan nakayama type rule a frobenius type formula and an analogue of the determinantal formula for the generalized characters of s n

a frobenius formula for the characters of the hecke algebras this paper uses the theory of quantum groups and the quantum yang baxter equation as a guide in order to produce a method of computing the irreducible characters of the hecke algebra this approach is motivated by an observation of m jimbo giving a representation of the hecke algebra on tensor space which generates the full centralizer of a tensor power of the standard representation of the quantum group u q mathfrak s l n by rewriting the solutions of the quantum yang baxter equation for u q mathfrak s l n in a different form one can avoid the quantum group completely and produce a frobenius formula for the characters of the hecke algebra by elementary methods using this formula we derive a combinatorial rule for computing the irreducible characters of the hecke algebra this combinatorial rule is aq extension of the murnaghan nakayama for computing the irreducible characters of the symmetric group along the way one finds connections apparently unexplored between the irreducible characters of the hecke algebra and hall littlewood symmetric functions and kronecker products of symmetric groups

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

on cellular algebras with jucys murphy elements abstract we study analogues of jucys murphy elements in cellular algebras arising from repeated jones basic constructions examples include brauer and bmw algebras and their cyclotomic analogues

on brauer algebra simple modules over the complex field this paper gives two results on the simple modules for the brauer algebra bn over the complex field first we describe the module structure of the restriction of all simple bn modules to bn 1 second we give a new geometrical interpretation of ram and wenzl s construction of bases for permissible simple modules

representations and traces of the hecke algebras hn q of type an 1 the notion of the connectivity class of minimal words in the algebra hn q is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented guided by these results the connection between the ocneanu trace on hn q and schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of hn q they appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to schur functions their evaluation involves the use of the littlewood richardson rule which is proved to apply to hn q just as it does to sn both representation matrices and characters are tabulated

on central idempotents in the brauer algebra abstract we provide a method for constructing central idempotents in the brauer algebra using the splitting of short exact sequences of bimodules from this we determine certain primitive central idempotents by working over a suitable integral ring we hence demonstrate an efficient method of constructing pieces of the representation theory of the brauer algebra over artinian rings from the integral case

a recursive rule for kazhdan lusztig characters abstract a formula for computing kazhdan lusztig characters in hecke algebras of coxeter groups is given this formula holds for products of coxeter elements of parabolic subgroups in an arbitrary coxeter group where the parabolic subgroups are of type a and the kazhdan lusztig representations are of finite dimension the formula yields new combinatorial interpretations of murnaghan nakayama type to these characters and to other related characters

bethe subalgebras in affine birman murakami wenzl algebras and flat connections for q kz equations commutative sets of jucys murphy elements for affine braid groups of types were defined construction of r matrix representations of the affine braid group of type and its distinguished commutative subgroup generated by the type jucys murphy elements are given we describe a general method to produce flat connections for the two boundary quantum knizhnik zamolodchikov equations as necessary conditions for sklyanin s type transfer matrix associated with the two boundary multicomponent zamolodchikov algebra to be invariant under the action of the type jucys murphy elements we specify our general construction to the case of the birman murakami wenzl algebras bmw algebras for short as an application we suggest a baxterization of the dunkl cherednik elements in the double affine hecke algebra of type a

a classification of simple lie modules having a 1 dimensional weight space let l denote a simple lie algebra over the complex numbers in this paper we classify and construct all simple l modules which may be infinite dimensional but have at least one 1 dimensional weight space this completes the study begun earlier by the authors for the case of l a the approach presented here relies heavily on the results of suren fernando whose dissertation dealt with simple weight modules and their weight systems 0 introduction let l be a finite dimensional simple lie algebra over the complex field c having a cartan subalgebra h and denote by c l the centralizer of h in the universal enveloping algebra u of l if x h c is a weight function of a 1 dimensional weight space mx in a simple l module m then q c l c defined by q c v cv for v e mx and c e c l is an algebra homomorphism called a mass function of m clearly x restricted to h is equal to x conversely given any algebra homomorphism q c l c one can construct a unique simple l module which admits x as a mass function 4 10 in 4 the authors determined all algebra homomorphisms q c l c for the simple lie algebras l of type an and using these classified all pointed an modules where we call a module pointed if it is simple and has at least one 1 dimensional weight space in this paper we complete the classification of all pointed l modules for arbitrary simple lie algebras the collection of all pointed l modules clearly includes the highest weight l modules and is included in the collection of all harish chandra l modules relative to the cartan subalgebra h i e simple l modules having a weight space decomposition and finite dimensional weight spaces relative to h the latter inclusion is strict since there exist examples of harish chandra a2 modules in which every weight space is two dimensional in the special case of a1 modules every harish chandra a1 module is pointed our approach to this problem makes heavy use of the results from fernando s thesis 8 in particular from fernando s results we see that in place of determining all algebra homomorphisms q c l c it suffices to find only those algebra homomorphisms x which are associated with so called pointed torsion free l received by the editors may 23 1984 this paper was presented at the 1984 canadian mathematical society s summer seminar on lie algebras and related topics held at the university of windsor 1980 mathematics subject classification primary 17b10

on a family of non unitarizable ribbon categories we consider several families of categories the first are quotients of h andersen s tilting module categories for quantum groups of lie type b at odd roots of unity the second consists of categories of type bc constructed from idempotents in bmw algebras our main result is to show that these families coincide as braided tensor categories using a recent theorem of tuba and wenzl by appealing to similar results of blanchet and beliakova we obtain another interesting equivalence with these two families of categories and the quantum group categories of lie type c at odd roots of unity the morphism spaces in these categories can be equipped with a hermitian form and we are able to show that these categories are never unitary and no braided tensor category sharing the grothendieck semiring common to these families is unitarizable

strongly multiplicity free modules for lie algebras and quantum groups abstract let u be either the universal enveloping algebra of a complex semisimple lie algebra g or its drinfel d jimbo quantisation over the field c z of rational functions in the indeterminate z we define the notion of strongly multiplicity free smf for a finite dimensional u module v and prove that for such modules the endomorphism algebras end u v r are generic in the sense that in the classical unquantised case they are quotients of kohno s infinitesimal braid algebra t r while in the quantum case they are quotients of the group ring c z b r of the r string braid group b r in the classical case the generators are generalisations of the quadratic casimir operator c of u while in the quantum case they arise from r matrices which may be thought of as square roots of a quantum analogue of c in a completion of u r this unifies many known results and brings some new cases into their context these include the irreducible 7 dimensional module in type g 2 and arbitrary irreducibles for sl 2 the work leads naturally to questions concerning non semisimple deformations of the relevant endomorphism algebras which arise when the ground rings are varied

on the hecke algebras and the colored homfly polynomial the colored homfly polynomial is the quantum invariant of oriented links in s 3 associated with irreducible representations of the quantum group u q sl n in this paper using an approach to calculate quantum invariants of links via the cabling projection rule we derive a formula for the colored homfly polynomial in terms of the characters of the hecke algebras and schur polynomials the technique leads to a fairly simple formula for the colored homfly polynomial of torus links this formula allows us to test the labastida marino vafa conjecture which reveals a deep relationship between chern simons gauge theory and string theory on torus links

weight modules for current algebras for any finite dimensional simple lie algebra g and commutative associative algebra s of finite type we give a complete classification of the simple weight modules of g s with bounded weight multiplicities the classification of simple weight modules is widely seen as a difficult problem in a spectacular tour de force olivier mathieu gave a complete description of such modules for finite dimensional simple lie algebras ma the crucial part of mathieu s classifi cation was understanding the simple admissible highest weight modules in the present paper we consider this question in the context of generalised current algebras l g k s where g is a finite dimensional simple lie algebra over k and s is a commutative associative and unital k algebra which is of finite type these algebras include the finite dimensional simple lie algebras g and the loop algebras g c c t t 1 of affine kac moody theory our methods work not only for the classification of simple admissible highest weight modules but also describe the simple objects in the much larger category of admissible modules weight modules whose weight multiplicities are uniformly bounded we hope that this paper will contribute to solving the harder problem of classifying all simple weight modules for these algebras the category of admissible modules includes the finite dimensional simple modules all of which are known to be evaluation modules see the introductions of l1 cfk or the more general results in nss lp l2 for details what is surprising is that this larger category of generally infinite dimensional modules can also be described in terms of evaluations we now summarize the content of this paper funding from the natural sciences and engineering research council of canada is gratefully ac knowledged

subduction coefficients of birman wenzl algebras and racah coefficients of the quantum groups oq n and spq 2m i subduction coefficients irreducible representations of birman wenzl algebras cf r q in the non standard basis are discussed a procedure for evaluating subduction coefficients sdcs or the transformation coefficients between standard and non standard basis of birman wenzl algebras is formulated based on the linear equation method sdcs of cf r q with f 4 are derived racah coefficients of the quantum groups oq n and spq 2m can be obtained from sdcs of birman wenzl algebras cf r q by using the schur weyl brauer duality relation between birman wenzl algebras cf r q and the quantum groups oq n and spq 2m

murnaghan nakayama rules for characters of iwahori hecke algebras of classical type in this paper we give murnaghan nakayama type formulas for computing the irreducible characters of the iwahori hecke algebras of types an 1 bn and dn our method is a generalization of a derivation of the murnaghan nakayama formula for the irreducible characters of the symmetric group given by curtis greene greene s approach is to sum up the diagonal entries of the matrices of certain cycle permutations in young s seminormal representations the analogues of the young seminormal representations for the iwahori hecke algebras of types an 1 bn and dn were given by hoefsmit

frt duals as quantum enveloping algebras the hopf algebra generated by the l functionals on the quantum double cq g cq g is considered where cq g is the coordinate algebra of a standard quantum group and q is not a root of unity it is shown to be isomorphic to cq g op uq g this proves a conjecture by t hodges as an algebra it can be embedded into uq g uq g here it is proven that there is no bialgebra structure on uq g uq g for which this embedding becomes a homomorphism of bialgebras in particular it is not an isomorphism as a preliminary a lemma of hodges concerning the structure of l functionals on cq g is generalized for the classical groups a certain choice of root vectors is expressed in terms of l functionals a formula for their coproduct is derived