Benjamin Steinberg
We say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field $\Bbbk$.
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Affine and cyclotomic webs
本文引入了仿射web范畴及其循环商范畴，并建立了与有限$W$-代数的联系，从而得到了$W$-Schur代数幂等子代数的图示表示。
The character degree product and the conjugacy length product for finite general linear groups
本文针对 K. Harada 关于有限群不可约特征标次数乘积整除共轭类长度乘积的猜想，在有限一般线性群与有限酉群上完成了验证。
We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GL_n(k) for n < 12 and k a perfect field. If our field is finite of …
Type AIII orbits in the affine flag variety of type A
本文研究了类型AIII轨道在仿射旗形 variety 中的参数化问题，通过构造显式的双射关系，将轨道与仿射(p,q)-clans联系起来。
Vanishing pairs of conjugacy classes for the symmetric group
本文刻画了对称群中满足任一非线性不可约特征标至少在其一上取零的共轭类对。
Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the general linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$ of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a function of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomial in $q$ with integer coefficients.
Irreducible representations of conjugacy quandles
研究共轭quandle的不可约表示，通过群的特征标与线性表示的乘积来描述，证明其 enveloping group 的嵌入性质及完美群时的同构性。
We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth po…
Conjugacy of Isometries in Real Orthogonal Groups
研究实正交群中等距的共轭性质，确定所有正交变换，使得线性群中与之共轭的变换在正交群中也共轭。
Let $\mathsf G$ be a connected reductive linear algebraic group defined over $\mathbb R$, and let $C: \mathsf G\rightarrow \mathsf G$ be a fundamental Chevalley involution. We show that for every $g\in \mathsf G(\mathbb R)$, $C(g)$ is conjugate to $g^{-1}$ in the group $\mathsf G(\mathbb R)$. Similar result on the Lie algebras is also obtained.
Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e. that conjugacy classes of S_n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S_n x S_n. We also consider the permutation representations of S_n which arise from the action of S_n on k-tuples, and classify which of them unite conjugacy classes and wh…
The conjugacy action of $S_n$ and modules induced from centralisers
本研究证明对称群特征标表中不可约特征标对应的行和在共轭类子集上的正性，构建了分析共轭作用相关诱导表示的通用框架，并发现了新的含所有不可约表示的$n!$维扭共轭表示。
Kronecker Coefficients and Simultaneous Conjugacy Classes
该研究引入克罗内克赫克代数研究有限群的克罗内克系数，建立同时共轭类数量与克罗内克系数的关系，定义双实群并刻画其与无重数张量积的关联，证明多类有限群不具有无重数张量积。
Let f(x)=Ax+b and g(x)=Cx+d be two affine operators given by n-by-n matrices A and C and vectors b and d over a field F. They are said to be biregularly conjugate if hf=gh for some bijection h: F^n-->F^n being biregular, this means that the coordinate functions of h and h^{-1} are polynomials. Over an algebraically closed field of characteristic 0, we obtain necessary and sufficient conditions of biregular conjugacy of affine operators and give a canonical form of an affine operator up to biregular conjugacy. These results for bijective affine operators were obtained by J.Blanc [Conjugacy clas…
On common zeros of characters of finite groups
本文研究有限群的零化元素与零化共轭类问题，重点探讨了使所有非线性不可约复特征标均在其上取值为零的最小共轭类数量，以及存在公共零点的非线性不可约特征标的最小数量。
Let W be an Iwahori-Weyl group of a connected reductive group G over a non-archimedean local field. I prove that if w is an element of W that does not act on the corresponding apartment of G by a translation then one can apply to w a sequence of conjugations by simple reflections, each of which is length-preserving, resulting in an element w' for which there exists a simple reflection s such that l(sw's)>l(w'). Even for affine Weyl groups, a special case of Iwahori-Weyl groups and also an important subclass of Coxeter groups, this is a new fact about conjugacy classes. Further, there are impli…
When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the values of irreducible characters of the extended groups on these elements. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.
Let G be a noncyclic group of order 4, and let K be the ring Z of rational integers, the localization of Z at the prime 2 and the ring of 2-adic integers, respectively. We describe, up to conjugacy, all of the indecomposable subgroups in the group GL(m,K) which are isomorphic to G.
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles …
Finite-dimensional modules over associative equivariant map algebras
本文研究线性约化群作用下结合等变映射代数的有限维表示，通过闭轨道点的迷向群不动点子代数刻画单模，并给出整个范畴按闭轨道的直和分解。
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily and we show that it very closely related to the combinatorial approach to random matrices. Our formulas are exact (in a sense that they hold not only asymptotically for large q). This result has many inte…
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