Bitableaux Bases for some Garsia-Haiman Modules and Other Related Modules

  • E. E. Allen


For certain subsets $S$ and $T$ of $${\cal A} =\bigl\{\cdots, (0,2), (0,1), (0,0), (1,0), (2,0), \cdots \bigr\}$$ and factor spaces ${\bf C}_{S}[X,Y]$, ${\bf C}^{+}_{S,T}[X,Y,Z,W]$ and ${\bf C}^{-}_{S,T}[X,Y,Z,W]$, bitableaux bases are constructed that are indexed by pairs of standard tableaux and sequences in the collections $\Upsilon_{\psi_S}$ and $\Upsilon_{\psi_T}$. These bases give combinatorial interpretations to the appropriate Hilbert series of these spaces as well as the graded character of ${\bf C}_{S}[X,Y]$.

The factor space ${\bf C}_{S}[X,Y]$ is an analogue of the coinvariant ring of a polynomial ring in two sets of variables. ${\bf C}^{+}_{S,T}[X,Y,Z,W]$ and ${\bf C}^{-}_{S,T}[X,Y,Z,W]$ are analogues of coinvariant spaces in symmetric and skew-symmetric polynomial settings, respectively. The elements of the bitableaux bases are appropriately defined images in the polynomial spaces of bipermanents. The combinatorial interpretations of the respective Hilbert series and graded characters are given by statistics based on cocharge tableaux. Additionally, it is shown that the Hilbert series and graded characters factor nicely. One of these factors gives the Hilbert series of a collection of Schur functions $s_{\lambda/\mu}$ where $\mu$ varies in an appropriately defined $\lambda$.

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