Calibrated representations of the double Dyck path algebra

IF 1.3 2区 数学 Q1 MATHEMATICS Mathematische Annalen Pub Date : 2024-08-06 DOI:10.1007/s00208-024-02937-2
Nicolle González, Eugene Gorsky, José Simental
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Abstract

The double Dyck path algebra \(\mathbb {A}_{q,t}\) and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra \(\mathbb {B}_{q,t}\) was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra \(\mathbb {B}_{q,t}\). We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces.

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双戴克路径代数的校准表示法
双戴克路径代数(\mathbb {A}_{q,t}\ )及其多项式表示最初是作为卡尔松和梅利特著名的洗牌定理证明的关键人物出现的。随后,第二位作者以及卡尔松和梅利特利用抛物线旗希尔伯特方案的 K 理论给出了等价代数 \(\mathbb {B}_{q,t}\) 的几何表述。在本文中,我们开始系统地研究双戴克路径代数 \(\mathbb {B}_{q,t}\) 的表示理论。我们定义了这个代数的自然扩展,并研究了它的校准表示。我们证明多项式表示是校准表示,并把它归入由满足特定条件的正集构造的校准表示大家族。我们还定义了这些表示的张量乘积和对偶,从而证明(在合适的条件下)校准表示的范畴一般是单义的。作为应用,我们证明了多项式表示的张量幂可以从抛物线吉塞克模空间的等变 K 理论中构造出来。
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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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