I am not so sure that if you are explaining the definition that you want to state it in terms of "the multiplicity of the trivial..." It kind of depends on where you put this method. One place to consider is in the Partition class or in PartitionTuple class.
For example: PartitionTuple([[3,1],[2,2],[3,1]]).kronecker_coefficient() or Partition([3,1]).kronecker_coefficient([2,2],[3,1]) You might check if this procedure is faster (however you should note that it is called for example kronecker_coefficient([3,1],[2,2],[3,1])): sage: from sage.libs.symmetrica.symmetrica import charvalue_symmetrica as chv sage: def kronecker_coefficient(*partns): ....: if partns==(): ....: return 1 ....: else: ....: return sum(mul(chv(la,mu) for la in partns) for mu in Partitions(sum(partns[0]))) ....: This version automatically has an optional last parameter. -Mike On Monday, 14 November 2016 05:53:10 UTC-5, Dima Pasechnik wrote: > > > > On Monday, November 14, 2016 at 10:44:35 AM UTC, Amri wrote: >> >> But you just add the "arbitrary irreducible" to your list: >> >> mult of la in mu\otimes nu = mult of triv in mu\otimes\nu\otimes la >> >> there is complete symmetry between the parameters. >> > > Well, this still does not imply that one doesn't want la as an extra > parameter for > the function, even if only for convenience. > (imagine a user that has a bit less representation theory of S_n loaded in > his/her head > than you :-)) > > >> On Monday, November 14, 2016 at 3:54:24 PM UTC+5:30, Dima Pasechnik wrote: >>> >>> >>> >>> On Monday, November 14, 2016 at 5:36:19 AM UTC, Amri wrote: >>>> >>>> It would be nice to have a function for kronecker coefficients in sage. >>>> I would be happy to implement it once I have some inputs on what would be >>>> the best way/place. One definition of the kronecker coefficient is the >>>> following: >>>> >>>> Defn. Given a sequence [la1, ...., lan] of ineteger partitions on n, >>>> the kronecker coef of this sequence is the multiplicity of the trivial >>>> representation the tensor product of the Specht modules corresponding to >>>> the partitions la1, la2,...,lan. >>>> >>> >>> I'd say that there should be a function for the multiplicity of an >>> arbitrary irreducible, not only the trivial one. >>> >>> >>>> >>>> Note: some people like to think of it slightly differently: given three >>>> partitions la, mu and nu of n, the kronecker coefficient is the >>>> multiplicity of V_la in V_mu \times V_nu. This coincided with the >>>> multiplicity of the trivial rep in V_la\otimes V_\mu\otimes V_\nu (which >>>> is >>>> the definition above). >>>> >>>> A quick implementation is: >>>> sage: def kronecker_coefficient(partns): >>>> ....: S = SymmetricFunctions(QQ).schur() >>>> ....: pr = reduce(lambda x, y: x.itensor(y), [S[la] for la in >>>> partns]) >>>> ....: return pr.coefficient(Partition([sum(partns[0])])) >>>> >>>> This code is already in the documentation for a patch of mine at: >>>> >>>> https://trac.sagemath.org/ticket/17437 >>>> >>>> Looking forward to some feedback. >>>> >>>> Thanks, >>>> Amri. >>>> >>> -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sage-combinat-devel. For more options, visit https://groups.google.com/d/optout.
