Higher rank (𝑞,𝑡)-Catalan polynomials, affine Springer fibers, and a finite rational shuffle theorem

IF 1.2 2区数学 Q1 MATHEMATICS

Transactions of the American Mathematical Society Pub Date : 2024-01-03 DOI:10.1090/tran/9115

Nicolle González, José Simental, Monica Vazirani

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引用次数: 0

Abstract

We introduce the higher rank ( q , t ) (q,t) -Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a d i n v \mathtt {dinv} statistic on rank r r semistandard ( m , n ) (m,n) -parking functions and prove c o d i n v \mathtt {codinv} counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Catalan numbers in the non-coprime case.

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高阶(△,𝑡)-卡塔兰多项式、仿射 Springer 纤维和有限有理洗牌定理

我们引入了高阶 ( q , t ) (q,t) - 卡塔兰多项式，并证明它们等于疋田多项式对有限变量的截断。利用仿射组合和某种标准化映射，我们定义了一个关于秩 r r 半标准 ( m , n ) (m,n) -停车函数的 d i n v \mathtt {dinv}统计量，并证明了 c o d i n v \mathtt {codinv} 在抛物线仿射 Springer 纤维的仿射铺设中计算仿射空间的维数。结合这些结果，我们给出了双仿射赫克代数中有理洗牌定理的有限类比。最后，我们还给出了非幂情况下高阶加泰罗尼亚数的比兹利式公式。

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来源期刊

Transactions of the American Mathematical Society 数学-数学

CiteScore

2.30

自引率

7.70%

发文量

171

审稿时长

3-6 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles in all areas of pure and applied mathematics. To be published in the Transactions, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Papers of less than 15 printed pages that meet the above criteria should be submitted to the Proceedings of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.