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Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials) | | Robert A. Proctor
; Matthew J. Willis
; | | Date: |
14 Jun 2017 | | Abstract: | Fix an integer partition lambda that has no more than n parts. Let beta be a
weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur
function indexed by lambda and beta is a polynomial generating function in x_1,
.., x_n for certain semistandard tableaux of shape lambda. Let pi be an
n-permutation. The type A Demazure character (key polynomial, Demazure
polynomial) indexed by lambda and pi is another such polynomial generating
function. Reiner and Shimozono and then Postnikov and Stanley studied
coincidences between these two families of polynomials. Here their results are
sharpened by the specification of unique representatives for the equivalence
classes of indexes for both families of polynomials, extended by the
consideration of more general beta, and deepened by proving that the polynomial
coincidences also hold at the level of the underlying tableau sets. Let R be
the set of lengths of columns in the shape of lambda that are less than n.
Ordered set partitions of {1,..,n} with block sizes determined by R, called
R-permutations, are used to describe the minimal length representatives for the
parabolic quotient of the nth symmetric group specified by the set
{1,..,n-1}R. The notion of 312-avoidance is generalized from n-permutations to
these set partitions. The R-parabolic Catalan number is defined to be the
number of these. Every flagged Schur function arises as a Demazure polynomial.
Those Demazure polynomials are precisely indexed by the R-312-avoiding
R-permutations. Hence the number of flagged Schur functions that are distinct
as polynomials is shown to be the R-parabolic Catalan number. The projecting
and lifting processes that relate the notions of 312-avoidance and of
R-312-avoidance are described with maps developed for other purposes. | | Source: | arXiv, 1706.4649 | | Services: | Forum | Review | PDF | Favorites | |
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