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A Proof of then! Conjecture for Generalized Hooks

✍ Scribed by Ethan Reiner

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 614 KB
Volume: 75
Category: Article
ISSN: 0097-3165
DOI: 10.1006/jcta.1996.0060

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✦ Synopsis

In [4], Garsia and Haiman [Electronic J. of Combinatorics 3, No. 2 (1996)] pose a conjecture central to their study of the Macdonald polynomials H + (x; q, t). For each + | &n one defines a certain determinant 2 + (X n , Y n ) in two sets of variables. The n! conjecture asserts that the vector space given by the linear span of derivatives of 2 + , written L[ p x q y 2 + (X n , Y n )], has dimension n!. The conjecture reduces to a well-known result about the Vandermonde determinant when +=(1 n ) or +=(n) (see [1], for example). Garsia and Haiman (see [Proc. Natl. Acad. Sci. 90 (1993), 3607 3610]) have demonstrated the conjecture for two-rowed shapes +=(a, b), two-columned shapes +=(2 a , 1 b ) and hook shapes +=(a, 1 b ). In this paper, we give an overview of the methods used by Reiner in his thesis to prove the n! conjecture for generalized hooks, that is, for +=(a, 2, 1 b ).

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