Christine Bessenrodt, Thorsten Holm, Alexander Zimmermann
We show how to extend the theory of generalized Reynolds ideals, as introduced by B. K"ulshammer, from symmetric algebras to arbitrary finite-dimensional algebras (in positive characteristic). This provides new invariants of the derived module categories of finite-dimensional algebras, extending recent results by the third author for symmetric algebras.
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Generalized Reynolds ideals are ideals of the center of a symmetric algebra over a field of positive characteristic. They have been shown by the second author to be invariant under derived equivalences. In this paper we determine the generalized Reynolds ideals of algebras of dihedral and semidihedral type (as defined by Erdmann), in characteristic 2. In this way we solve some open problems about scalars occurring in the derived equivalence classification of these algebras.
For any algebraically closed field $k$ of positive characteristic $p$ and any non negative integer $n$ Külshammer defined ideals $T\_nA^\perp$ of the centre of a symmetric $k$-algebra $A$. We show that for derived equivalent algebras $A$ and $B$ there is an isomorphism of the centres of $A$ and $B$ mapping $T\_nA^\perp$ to $T\_nB^\perp$ for all $n$. Recently Héthelyi, Horváth, Külshammer and Murray showed that this holds for Morita equivalent algebras.
We study symmetric algebras $A$ over an algebraically closed field $F$ in which the Jacobson radical of the center of $A$, the socle of the center of $A$ or the Reynolds ideal of $A$ are ideals.
Greene's Theorem and ideals of the group algebra of a symmetric group
本文利用Greene定理研究对称群群代数，证明相关商环具有群元素自然基，并给出特定置换模零化子的生成元。
Let $F$ be a field of characteristic $p > 0$. We study the structure of the finite groups $G$ for which the socle of the center of $FG$ is an ideal in $FG$ and classify the finite $p$-groups $G$ with this property. Moreover, we give an explicit description of the finite groups $G$ for which the Reynolds ideal of $FG$ is an ideal in $FG$.
We give a complete derived equivalence classification of all symmetric algebras of domestic representation type over an algebraically closed field. This completes previous work by R. Bocian and the authors, where in this paper we solve the crucial problem of distinguishing standard and nonstandard algebras up to derived equivalence. Our main tool are generalized Reynolds ideals, introduced by B. K"ulshammer for symmetric algebras in positive characteristic, and recently shown by A. Zimmermann (math.RA/0509580) to be invariants under derived equivalences.
If $Λ$ is an indecomposable, non maximal, symmetric order, then the idealizer of the radical $Γ:= \Id(J(Λ)) = J(Λ)^{#} $ is the dual of the radical. If $Γ$ is hereditary then $Λ$ has a Brauer tree (under modest additional assumptions). Otherwise $Δ:= \Id(J(Γ)) = (J(Γ)^2)^{#} $. If $Λ= \Z_p G$ for a $p$-group $G\neq 1$, then $Γ$ is hereditary iff $G\cong C_p$ and otherwise $[Δ: Λ] = p^2 | G/(G'G^p)| $. For Abelian groups $G$, the length of the radical idealizer chain of $\Z_pG$ is $(n-a)(p^{a} - p^{a-1})+p^{a-1}$, where $p^n$ is the order and $p^a$ the exponent of the Sylow $p$-subgroup of $G$.
A special class of prime ideals for infinite symmetric group algebras
本文研究了无限对称群代数中一类特殊素理想，揭示其半环结构，并在复数域上与球形表示建立联系，在正特征下探讨其与对称张量范畴结构的关联。
An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of …
Maximal ideals in the finitary symmetric group algebra in characteristic two
本文解决了特征2域上有限对称群代数极大理想分类的悬而未决问题，通过关联其素理想与高阶Verlinde范畴，借助研究一般线性群倾斜模的厚张量理想完成了该工作。
Symmetric and Isotypic Hilbert Series for Symmetric Ideals
本文定义并研究了可数变量多项式环中对称理想的对称和等型Hilbert级数，证明了在温和支撑条件下对称Hilbert级数为有理函数，且该有理性质可推广到每个不可约表示的等型Hilbert级数。
Let A be a finite dimensional symmetric cllular algebras. We construct a nilpotent ideal in A. The ideal connects the radicals of cell modules with the radical of the algebra. It also reveals some information on the dimensions of simple modules of A.
In by now classical work, K. Erdmann classified blocks of finite groups with dihedral defect groups (and more generally algebras of dihedral type) up to Morita equivalence. In the explicit description by quivers and relations of such algebras with two simple modules, several subtle problems about scalars occurring in relations remained unresolved. In particular, for the dihedral case it is a longstanding open question whether blocks of finite groups can occur for both possible scalars 0 and 1. In this article, using Kuelshammer ideals (a.k.a. generalized Reynolds ideals), we provide the first …
Resolutions of symmetric ideals via stratifications of derived categories
提出通过对相关导出范畴分层的方法统一多项式环中对称理想的稳定性结果，证明了特定序列的猜想，并获得了有界导出范畴的半正交分解。
We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these ideals, and describe formulas for the graded characters of the corresponding quotient rings.
We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three - with one exception, the secant variety of the $21$-dimensional spinor variety in $\pp{63}$ where we show the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.
Let $R$ be an integral domain and $A$ a symmetric cellular algebra over $R$ with a cellular basis $\{C_{S,T}^\lam \mid \lam\inΛ, S,T\in M(\lam)\}$. We will construct an ideal $L(A)$ of the center of $A$ and prove that $L(A)$ contains the so-called Higman ideal. When $R$ is a field, we prove that the dimension of $L(A)$ is not less than the number of non-isomorphic simple $A$-modules.
For a perfect field $k$ of characteristic $p>0$ and for a finite dimensional symmetric $k$-algebra $A$ Külshammer studied a sequence of ideals of the centre of $A$ using the $p$-power map on degree 0 Hochschild homology. In joint work with Bessenrodt and Holm we removed the condition to be symmetric by passing through the trivial extension algebra. If $A$ is symmetric then the dual to the Külshammer ideal structure was generalised to higher Hochschild homology in earlier work. In the present paper we follow this program and propose an analogue of the dual to the Külshammer ideal structure on t…
Given a collection of $t$ subspaces in an $n$-dimensional $\mathbb{K} $-vector space $W$ we can associate to them $t$ vanishing ideals in the symmetric algebra $\mathcal{S}(W^*) = \mathbb{K}[x_1,x_2,\dots,x_n]$. As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal. Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of $t$ linear ideals is equal to $t$. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of $t$ linear ideals is at most $t$ and they also showed…
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