On the Brauer monoid for finite fields
V. V. Kirichenko, B. V. Novikov
利用0-上同调的概念研究Brauer么半群。我们研究了可逆元素的修改对Brauer幺半群结构的影响，特别是对于有限域。
The Brauer monoid is studied by the notion of 0-cohomology. We investigate the impact of invertible elements of modifications on the structure of the Brauer monoid, especially for finite fields.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
On presentations of Brauer-type monoids
我们得到介绍的布劳尔幺半群，部分类似的布劳尔幺半群，并为最大的可因式分解逆子幺半群的对偶对称逆幺半群。在所有三种情况下，我们应用相同的方法，基于所有这些幺半群作为Brauer型幺半群的实现。
Presentation of the singular part of the Brauer monoid
我们得到了关于一个由幂等元组成的不可约生成元系统的Brauer么半群的奇异部分的一个表示。作为这一结果的应用，我们得到了一个利用子集的连通序列构造对称群的新方法。另一个应用描述了在Brauer么半群的奇异部分中的元素的长度相对于上面提到的生成元系统。
We show that it is co-NP-hard to check whether a given semigroup identity holds in the twisted Brauer monoid $\mathcal{B}^τ_n$ with $n\ge5$.
We study monoids generated by various combinations of idempotents and one- or two-sided units of an infinite partial Brauer monoid. This yields a total of eight such monoids, each with a natural characterisation in terms of relationships between parameters associated to Brauer graphs. We calculate the relative ranks of each monoid modulo any other such monoid it may contain, and then apply these results to determine the Sierpinski rank of each monoid, and ascertain which ones have the semigroup Bergman property. We also make some fundamental observations about idempotents and units in arbitrar…
A variant of a semigroup $S$ with respect to an element $a\in S$ is the semigroup $S^{a}=(S,\star_{a})$, where $x\star_{a}y=xay$ for any $x,y\in S$. Here, $a$ is the sandwich element of $S^{a}$. In this article, we study variants of the partial Brauer monoid $\mathcal{PB}_{n}$ for $n\in\mathbb{N}$. We give the classification of these variants in the case when the rank of the sandwich element is nonzero.
We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated su…
We investigate the structure of the twisted Brauer monoid $\mathcal B_n^τ$, comparing and contrasting it to the structure of the (untwisted) Brauer monoid $\mathcal B_n$. We characterise Green's relations and pre-orders on $\mathcal B_n^τ$, describe the lattice of ideals, and give necessary and sufficient conditions for an ideal to be idempotent-generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent-generated, its…
We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate a…
We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I->M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.
Involutions of (twisted) diagram monoids
研究对划分、平面划分等多种图幺半群及其扭曲图幺半群的对合进行分类，刻画产生星正则或正则星幺半群结构的对合，发展扭曲积对合理论，给出相关结果如布劳尔幺半群的特殊情况及整数上扭曲图幺半群的性质等。
We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various $\mathscr R$-, $\mathscr L$- and $\mathscr D$-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.
A well-known theorem of Burnside says that if $ρ$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $ρ$. In 1962, R. Steinberg gave a module theoretic proof that simultaneously removed the constraint on the characteristic, and allowed the group to be replaced by a monoid. Brauer subsequently simplified Burnside's proof and, moreover, showed that if the character of $ρ$ takes on $r$ distinct values, then the first $r$ tensor powers of $ρ$ already contain amongst them a…
Affine diagram categories, algebras and monoids
本文研究了仿射图代数的生成元和关系，探讨了其表示论及张量积表示的渐进行为，并验证了一个关于有限单体表示中不可分解因子数量的猜想。
Tensor powers of representations of (diagram) monoids
本文研究有限幺半群表示的张量幂，聚焦合成长度与不可分解项的增长，对Temperley-Lieb、Motzkin和Brauer等图幺半群计算显式数据并分析分解模式。
We describe and count the maximal subsemigroups of many well-known monoids of transformations and monoids of partitions. More precisely, we find the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by V. H. Fernandes and co-authors (12 monoids in total); the partition, Brauer, Jones, and Motzkin monoids; and certain further monoids. Although descriptions of the maximal subsemigroups of some of the aforementioned classes of monoids appear in the literature, we present a unified framework for determining…
Finite groups with exactly two nonlinear irreducible $p$-Brauer characters
本文分类了恰有两个非线性不可约 $p$-Brauer 特征的有限群，并指出阶不被 $p$ 整除的情形已由 Pálfy 于 1981 年解决。
In this article we calculate two aspects of the representation theory of a Brauer configuration algebra: its Cartan matrix, and the module length of its associated indecomposable projective modules. Then we introduce the concept of subgroup-occurrence of an element in a group and use the previous aspects to demonstrate combinatorial equalities satisfied for any finite group.
If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise to isomorphic permutation representations C[X] and C[Y]. Equivalently, the map from the Burnside ring to the representation ring of G has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave-Bouc classification in the case of p-groups.
A generalization of the Brauer-Fowler theorem
本文利用有限单群分类推广了Brauer-Fowler定理，证明简单局部有限群若存在至多与$n$个对合交换的对合则必为有限且阶仅由$n$界定，回答了Strunkov的问题。
In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author. As a consequence, the Brauer trees of principal $\ell$-blocks of finite groups are known for $\ell>71$.
Brauer relations in finite groups II - quasi-elementary groups of order p^aq
这是第二次在一系列文件调查空间的布劳尔关系的有限群，内核的自然映射从其伯恩赛德环的有理表示环。第一个文件分类所有原始布劳尔关系，这是那些不来自适当的subjectients。在准初等群的情况下，描述是复杂的，它不指定在生成元和关系方面具有本原关系的群。在本文中，我们这样做的拟初等群的顺序p^aq。
In this paper, we study some variations of the well-known Itô-Michler theorem for $p$-Brauer characters using various inequalities involving the $p$-Brauer character degrees of finite groups. Several new criteria for the existence of a normal Sylow $p$-subgroup of finite groups are obtained using the $p$-parts and $p'$-parts of the $p$-Brauer character degrees.
Let $G$ be a finite solvable group, and let $p$ be a prime. In this note, we prove that $p$ does not divide $φ(1)$ for every irreducible monomial $p$-Brauer character $φ$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.
Let $G$ be a finite group, $p$ a prime, and $IBr_p(G)$ the set of irreducible $p$-Brauer characters of $G$. Let $\bar e_p(G)$ be the largest integer such that $p^{\bar e_p(G)}$ divides $χ(1)$ for some $χ\in IBr_p(G)$. We show that $|G:O_p(G)|_p \leq p^{k \bar e_p(G)}$ for an explicitly given constant $k$. We also study the analogous problem for the $p$-parts of the conjugacy class sizes of $p$-regular elements of finite groups.
We have found a list of finite simple groups with cyclic Sylow $p$-subgroup whose principal $p$-blocks have Brauer trees in the shape of a star, that is a tree of diameter at most $2$. Moreover, for an arbitrary finite group $G$ with cyclic Sylow $p$-subgroup, we have obtained a necessary condition when the Brauer tree of the principal $p$-block of $G$ is a star.
A local conjecture on Brauer character degrees of finite groups
最近，第二作者提出了关于有限群的不可约Brauer特征标的次数的一个新猜想。本文对有限群的块B提出了这个猜想的一个“局部”版本，给出了属于块B的不可约Brauer特征标的最大度的一个下界，它是关于块B的维数和不可约Brauer特征标的不变量，如亏损和不可约Brauer特征标的个数.我们还提出了一个较弱的版本，这个猜想轻微的重新制定导致有趣的公开问题的痕迹块的嘉当矩阵。然后，我们证明了强猜想是真实的块与一个简单的模块，块的p-可解群和块与循环亏损群。它也适用于零星群、对称群或李型群的块的许多进一步的例子。我们还表明，弱猜想是真实的驯服表示类型的块。
Unramified Brauer groups of finite and infinite groups
Bogomolov乘子是同构于给定商空间的非分歧Brauer群的群理论不变量。我们推导出一个同调版本的Bogomolov乘数，证明了一个霍普夫型公式，找到一个五项精确序列对应于这个不变量，并描述了中心扩展理论中的Bogomolov乘数的作用。给出了第二类幂零群的Bogomolov乘子的一个新刻画.我们定义的Bogomolov乘数在K-理论和证明其平凡性是等价于解决一个长期存在的问题所造成的低音。提出了一种计算Bogomolov乘子的算法。
In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely on the recent reduction of Willems' conjecture to a question on quasi-simple groups by Tong-Viet. We also verify the conditions of Tong-Viet for certain families of finite quasi-simple groups and odd primes. On the way we obtain lower bounds for the number of regular semisimple conjugacy classes in finite groups of Lie type.
We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets all the conjugacy classes of $p$-regular elements of $G$.
The Brauer relations of a finite group $G$ are virtual differences of non-isomorphic $G$-sets $X-Y$ which induce isomorphic permutation $G$-representations $\mathbb Q[X]\simeq\mathbb Q[Y]$ over the rationals. These relations have been classified by Tornehave-Bouc and Bartel-Dokchitser. Motivated by stable homotopy theory, a relative version of Brauer relations for $(G,C_p)$-bisets which are $C_p$-free have been classified by Kahn in case $G$ is an elementary Abelian $p$-group. In this paper we extend Kahn's classification to the case when $G$ is a finite Abelian $p$-group.
Finite semigroups embed in finitely presented congruence-free monoids
本文证明了有限半群嵌入到一个无同余幺半群中，并围绕Boone-Higman猜想提出了一些问题。
Brauer Theory for Profinite Groups
有限群的Brauer理论可以看作是比较群的特征为0的表示与群的素特征的表示的一种方法。在这里，我们推广了布劳尔理论的大部分机械设置的投射有限群。通过将Grothendieck群看作函子，我们刻画了投射有限群的相应Grothendieck群，并推广了分解映射，将其看作一个自然变换.讨论了投射有限群的特征标和Brauer特征标.给出了投射有限群的块理论的函子描述。最后给出了有限群$G$的Cartan矩阵的计算方法，给出了$G$与正规$p$ -子群的商的Cartan矩阵。
Graph products of residually finite monoids are residually finite
本文证明了剩余有限幺半群的图积是剩余有限的，并得出其自由积及半群图积的相应结论。
The simplicial complex of Brauer pairs of a finite reductive group
本文研究有限约化群的Brauer对偏序集诱导的单纯复形，明确其在定义特征下与Tits building同伦型一致，在非定义特征下与广义Harish-Chandra理论确定的复形同伦等价，拓展了Brown复形相关研究。
Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability
针对具有公共中心 p-子群的有限群，本文提出构造 Morita 型相对 Z-稳定等价的方法，推广 Linckelmann 的相关结论，并引入相对 Brauer 不可分解性概念及其等价判定条件。
Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer charact…
The Brauer group of modified supergroup algebras
计算了修正超群代数的Brauer群BM，特别是当基域代数闭且特征为零时，得到了所有有限维三角霍普夫代数的Brauer群BM的计算.并将所得结果与懒惰上同调的计算和张荫火的正合列的计算进行了比较。作为一个例子，我们以不可约根系的外尔群的扩张为群基准，以它们的标准表示为表示基准，显式地计算了修正超群代数的Brauer群和懒惰上同调.
A monoid $M$ is called surjunctive if every injective cellular automata with finite alphabet over $M$ is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non…
It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius-Schur indicator 1 can also be expressed in group theoretical terms. We show that this can done by counting solutions of $g_1^2\ldots g_n^2=1$ with $g_1,\ldots,g_n\in G$.
Two finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。