Some Natural Bigraded S_n-Modules and q,t-Kostka Coefficients.
Some Natural Bigraded S_n-Modules and q,t-Kostka Coefficients.
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一些自然二阶 S_n-模数和 q,t-Kostka 系数。
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发表时间：
1996
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通讯作者：
M. Haiman
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作者：
Taylor Coefficients;A. Garsia;M. Haiman
对每个μ n构造了一个双分次Sn-模Hμ，并猜想它的Frobenius特征Cμ（x; q，t）得到Macdonald系数Kλμ（q，t）.更准确地说，我们猜想Cμ（x; q，t）在Schur基上的展开式得到系数Cλμ（q，t），它与Kλμ（q，t）有恒等式Cλμ（q，t）= Kλμ（q，1/t）t的关系.这一点的有效性将给出麦克唐纳基{Pμ（x; q，t）}μ的表示理论设置，并建立麦克唐纳猜想，即Kλμ（q，t）是具有正整数系数的多项式。空间Hμ定义为某个双齐次多项式<$μ（x，y）对变量x1，x2，. . .，xn，y1，y2，. . .，yn.根据我们猜想的正确性，Hμ必然有n！维度我们把后者称为n！猜想这里将讨论这个猜想的几种等价形式及其一些结果。特别地，我们导出多项式Cλμ（q，t）与系数K λμ（q，t）= Kλμ（q，1/t）t具有许多共同的基本性质。例如，我们证明了Cλμ（0，t）= K <$λμ（0，t），Cλμ（q，0）= K <$λμ（q，0），并证明了在n！猜想我们也必须有等式Cλμ（1，t）= K <$λμ（1，t）和Cλμ（q，1）= K <$λμ（q，1）。当λ或μ是一个钩子时，证明等式Cλ μ（q，t）= Kλμ（q，1/t）t成立。当μ是一个2行或2列划分时，以及当μ是一个增广钩子时，它也被证明了（见[9]）。引言在本文中，μ将是n的一个分区，μ′将表示其共轭。我们还将用它的费勒斯图来表示μ。作为惯例，对于μ =（μ1 ≥ μ2 ≥ · ·· ≥ μk > 0），我们令
We construct for each μ ` n a bigraded Sn-module Hμ and conjecture that its Frobenius characteristic Cμ(x; q, t) yields the Macdonald coefficients Kλμ(q, t). To be precise, we conjecture that the expansion of Cμ(x; q, t) in terms of the Schur basis yields coefficients Cλμ(q, t) which are related to the Kλμ(q, t) by the identity Cλμ(q, t) = Kλμ(q, 1/t)t. The validity of this would give a representation theoretical setting for the Macdonald basis {Pμ(x; q, t)}μ and establish the Macdonald conjecture that the Kλμ(q, t) are polynomials with positive integer coefficients. The space Hμ is defined as the linear span of derivatives of a certain bihomogeneous polynomial ∆μ(x, y) in the variables x1, x2, . . . , xn, y1, y2, . . . , yn. On the validity of our conjecture Hμ would necessarily have n! dimension. We refer to the latter assertion as the n!-conjecture. Several equivalent forms of this conjecture will be discussed here together with some of their consequences. In particular, we derive that the polynomials Cλμ(q, t) have a number of basic properties in common with the coefficients K̃λμ(q, t) = Kλμ(q, 1/t)t. For instance, we show that Cλμ(0, t) = K̃λμ(0, t), Cλμ(q, 0) = K̃λμ(q, 0) and show that on the n! conjecture we must also have the equalities Cλμ(1, t) = K̃λμ(1, t) and Cλμ(q, 1) = K̃λμ(q, 1). The conjectured equality Cλμ(q, t) = Kλμ(q, 1/t)t will be shown here to hold true when λ or μ is a hook. It has also been shown (see [9]) when μ is a 2-row or 2-column partition and in [18] when μ is an augmented hook. Introduction Throughout this writing μ will be a partition of n and μ′ will denote its conjugate. We shall also identify μ with its Ferrers’ diagram. As customary, for μ = (μ1 ≥ μ2 ≥ · · · ≥ μk > 0), we let