Ribbon Schur functors

IF 0.9 1区 数学 Q2 MATHEMATICS
Keller VandeBogert
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引用次数: 0

Abstract

We investigate a generalization of the classical notion of a Schur functor associated to a ribbon diagram. These functors are defined with respect to an arbitrary algebra, and in the case that the underlying algebra is the symmetric/exterior algebra, we recover the classical definition of Schur/Weyl functors, respectively. In general, we construct a family of 3-term complexes categorifying the classical concatenation/near-concatenation identity for symmetric functions, and one of our main results is that the exactness of these 3-term complexes is equivalent to the Koszul property of the underlying algebra A. We further generalize these ribbon Schur functors to the notion of a multi-Schur functor and construct a canonical filtration of these objects whose associated graded pieces are described explicitly; one consequence of this filtration is a complete equivariant description of the syzygies of arbitrary Segre products of Koszul modules over the Segre product of Koszul algebras. Further applications to the equivariant structure of derived invariants, symmetric function identities, and Koszulness of certain classes of modules are explored at the end, along with a characteristic-free computation of the regularity of the Schur functor 𝕊λ applied to the tautological subbundle on projective space.

带舒尔函子
我们研究了与带状图相关的舒尔函子的经典概念的推广。这些函子是在任意代数上定义的,当底层代数是对称/外代数时,我们分别恢复了Schur/Weyl函子的经典定义。在一般情况下,我们构造了一组3项复形,对对称函数的经典串联/近串联恒等式进行了分类,我们的主要结果之一是这些3项复形的准确性相当于基础代数a的Koszul性质。我们进一步将这些带状舒尔函子推广到多舒尔函子的概念,并构造了一个正则过滤,其相关的分级块被明确描述;这种过滤的一个结果是对Koszul模的任意Segre积在Koszul代数的Segre积上的合的完全等变描述。进一步应用于派生不变量的等变结构,对称函数恒等式,以及某些模块类的Koszulness,以及在射影空间上应用于重言子束的Schur函子𝕊λ正则性的无特征计算。
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来源期刊
CiteScore
1.80
自引率
7.70%
发文量
52
审稿时长
6-12 weeks
期刊介绍: ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.
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