Schur-Weyl duality concerns the actions of $\text{GL}_{n}(\mathbb{C})$ and $S_{k}$ on tensor powers of the form $V^{\otimes k}$ for an $n$-dimensional vector space $V$. There are rich histories within representation theory, combinatorics, and statistical mechanics involving the study and use of diagram algebras, which arise through the restriction of the action of $\text{GL}_{n}(\mathbb{C})$ to subgroups of $\text{GL}_{n}(\mathbb{C})$. This leads us to consider further variants of Schur-Weyl duality, with the use of variants of the tensor space $V^{\otimes k}$. Instead of taking repeated tensor products of $V$, we make use of a freest commutative bilinear operation in place of $\otimes$, and this is motivated by an associated invariance property given by the action of $S_{k}$. By then taking the centralizer algebra with respect to the action of the group of permutation matrices in $\text{GL}_{n}(\mathbb{C})$, this gives rise to a diagram-like algebra spanned by a new class of combinatorial objects. We construct orbit-type bases for the centralizer algebras introduced in this paper, and we apply these bases to prove a combinatorial formula for the dimensions of our centralizer algebras.
The classical case of Schur--Weyl duality states that the actions of the group algebras of $GL_n$ and $S_d$ on the $d^{th}$-tensor power of a free module of finite rank centralize each other. We show that Schur--Weyl duality holds for commutative rings where enough scalars can be chosen whose non-zero differences are invertible. This implies all the known cases of Schur--Weyl duality so far. We also show that Schur--Weyl duality fails for $\mathbb{Z}$ and for any finite field when $d$ is sufficiently large.
Let $V$ be a free module of rank $n$ over a commutative unital ring $k$. We prove that tensor space $V^{\otimes r}$ satisfies Schur--Weyl duality, regarded as a bimodule for the action of the group algebra of the Weyl group of $\mathrm{GL}(V)$ and the partition algebra $P_r(n)$ over $k$. We also prove a similar result for the half partition algebra.
Finite Schur filtration dimension for modules over an algebra with Schur filtration
设G是GL_N或SL_N，G是正特征p的域k上的约化线性代数群，我们证明了仅当N2 ^N时的几个结果< 6 or p >.设G有理地作用在一个n-生成的交换k-代数A上。假设A作为G-模有一个好滤子或Schur滤子。设M是具有相容G作用的Noether A-模。则M有有限的good/Schur滤维数，使得至多有n个非零H^i（G，M）。此外，这些H^i（G，M）是不变量环A^G上的Noether模。我们的主要工具是一个决议涉及舒尔函子的理想的对角线的产品格拉斯曼。
We show that combinatorial objects called row-strict composition tableaux, introduced by Mason and Remmel in 2014 and closely related to the quasi-symmetric Schur functions of Haglund-Luoto-Mason-van Willigenburg, form a basis for Schur functors of finite free modules over arbitrary commutative rings. When the ring is the complex numbers, this produces a new basis for the irreducible polynomial representations of $\operatorname{GL}_n(\mathbb{C})$. Moreover, in this case it also produces new basis for the Plücker algebra, a subalgebra of the polynomial ring over $\mathbb{C}$ in $n^2$ variables…
The following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied to multiparameter persistent modules. The generic structure of such resolutions and the classifying spaces involved are studied using results spanning several decades of research in commutative algebra, beginning with the study of generic structural properties of free resolutions popularized by Buchsbaum and Eisenbud. Many explicit computations are presente…
Given an $\mathcal{N}=2$ superconformal field theory, we reconsider the Schur index $\mathcal{I}_L(q)$ in the presence of a half line defect $L$. Recently Cordova-Gaiotto-Shao found that $\mathcal{I}_L(q)$ admits an expansion in terms of characters of the chiral algebra $\mathcal{A}$ introduced by Beem et al., with simple coefficients $v_{L,β}(q)$. We report a puzzling new feature of this expansion: the $q \to 1$ limit of the coefficients $v_{L_,β}(q)$ is linearly related to the vacuum expectation values $\langle L \rangle$ in $U(1)_r$-invariant vacua of the theory compactified on $S^1$. This …
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.
