Hi Bruce, The research on these objects is still a bit ongoing so the conventions have not yet settled down which is why I hesitate to submit it to Sage until there is clean expository material to refer to. They will go into Sage eventually, but I think that they are closely related to plethysm, Kronecker and the restriction problem from Gl_n to S_n, so I think it is a very good idea to be careful about choosing the right notation and conventions.
Let me try to give you some references. For instance Halverson, Benkart and Harman wrote: https://arxiv.org/abs/1605.06543 see section 5. Do you see set valued tableaux? No, what you see is expressions involving set partitions and column strict tableaux (same difference if you know what you are looking at). You also have to know that the dimensions of these modules are the number of vascillating tableaux. However a year later Halverson and Benkart wrote some follow up expository work, they made the connection more explicit: https://arxiv.org/pdf/1709.07751.pdf And here they have the bijection between the vascillating tableaux and the set valued tableaux in Section 3.2. and the reason I am familiar with these connections is because Rosa Orellana and I wrote this paper: https://arxiv.org/abs/1605.06672 Section 6 gives the relationship between the partition algebra character values and the coefficients of the character basis in the power sum. The dimension (the number of vascillating tableaux of shape (n-|\lambda|,\lambda)) will be the coefficient of {\tilde s}_\lambda in p_{1^k} (and this is in Sage). For instance the example that I gave with the vascillating tableaux of size 2 (and n>4) are encoded in the coefficients of p_{11}: sage: SymmetricFunctions(QQ).inject_shorthands('all', verbose=False) sage: st(p[1,1]) 2*st[] + 3*st[1] + st[1, 1] + st[2] sage: s(p[1,1]).character_to_frobenius_image(5) s[3, 1, 1] + s[3, 2] + 3*s[4, 1] + 2*s[5] -Mike On Sunday, 20 May 2018 09:47:39 UTC-4, Bruce wrote: > > This is intriguing but I need some help as I am not familiar with these. > Can you give me some references on the background? > > In my setup, a "tableau" is a sequence of "partitions" with constraints > on neighbouring partitions. Do set valued tableaux, or multiset valued > tableaux, > fit this framework? > > P.S. Are you planning on submitting this code to sage? It seems that you > should, particularly as it is now in the public domain. > -- 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 sage-combinat-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-combinat-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-combinat-devel. For more options, visit https://groups.google.com/d/optout.
