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BGG and GLmnReps

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craicu edited this page May 13, 2016 · 3 revisions

Things we have done

BGG:

  • we have added the method bggComplex, which constructs (a truncation of) the linear complex over the polynomial ring (resp. exterior algebra) which corresponds via BGG to a graded module over the exterior algebra (resp. polynomial ring).

  • we have fixed the bug with directImageComplex pointed out here

##GLmnReps

We have started the package GLmnReps, which implements basic functionality to compute with representations of a general linear Lie superalgebra gl(m|n). The main application is to the study of syzygies of thickenings of determinantal varieties: we implement a conjectural description of the Betti tables of some of these thickenings.

  • bettiTableIlambda: let S denote the coordinate ring of the affine space of mxn matrices; given a partition lambda, let I_lambda denote the ideal generated by the irreducible GL_m x GL_n subrepresentation of S indexed by lambda. The method bettiTableIlambda computes the Betti table of I_lambda (using the recent speed-ups in version 1.9 for computing non-minimal resolutions in positive characteristic)

  • conjecturalBettiTableIlambda: this computes the Betti table of an ideal I_lambda, based on a conjectural description in terms of gl(m|n)-modules. It is significantly faster than bettiTableIlambda.

  • characterKlambda: given a partition lambda, the method computes the decomposition into irreducible representations of gl(m) x gl(n) for the Kac module K_lambda associated to lambda. As a module over the exterior algebra E, K_lambda is free, generated by the irreducible representation which is dual to that generating I_lambda.

  • characterLlambda: given a partition lambda, the method computes the decomposition into irreducible representations of gl(m) x gl(n) for the simple gl(m|n)-module L_lambda of lowest weight (lambda|lambda).

  • hilbertFunctionLlambda: considering L_lambda as a graded vector space, the method computes the dimensions of each of its graded components.

  • makeLinearComplexLGenericlambda: the method constructs the linear complex that corresponds via BGG to the module L_lambda, thought of as a module over the exterior algebra. The word ``Generic" in the name of the method is meant to indicate that the method only gives the right answer under certain genericity assumptions on lambda (for instance when lambda_i - lambda_(i+1) >= min(m,n)-1 for all i).

###Things we'd like to do

  • makeLinearComplexLlambda: a method that constructs the linear complex corresponding to L_lambda for an arbitrary lambda.

  • List the L_mu's that appear as composition factor of the Kac module K_lambda.

  • We are currently only dealing with modules of so-called maximal atypicality: instead we should consider arbitrary pairs of partitions (lambda,mu) and discuss more general Kac modules K_{lambda,mu} and simple modules L_{lambda,mu}.

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