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If $mathscr{J}$ is a finite-dimensional nilpotent algebra over a finite field $Bbbk$, the algebra group $P = 1+mathscr{J}$ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If $mathscr{J}$ is endowed with an involution $widehat{varsigma}$, then $widehat{varsigma}$ naturally defines a group automorphism of $P = 1+mathscr{J}$, and we may consider the fixed point subgroup $C_{P}(widehat{varsigma}) = {xin P : widehat{varsigma}(x) = x^{-1}}$. Assuming that $Bbbk$ has odd characteristic $p$, we use the standard supercharacter theory for $P$ to construct a supercharacter theory for $C_{P}(widehat{varsigma})$. In particular, we obtain a supercharacter theory for the Sylow $p$-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andre and Neto for the special case of the symplectic and orthogonal groups.
We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we pr
Let $L$ be a Lie algebra of Block type over $C$ with basis ${L_{alpha,i},|,alpha,iinZ}$ and brackets $[L_{alpha,i},L_{beta,j}]=(beta(i+1)-alpha(j+1))L_{alpha+beta,i+j}$. In this paper, we shall construct a formal distribution Lie algebra of $L$. Then
This paper reports some advances in the study of the symplectic blob algebra. We find a presentation for this algebra. We find a minimal poset for this as a quasi-hereditary algebra. We discuss how to reduce the number of parameters defining the alge
Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows t
In this joint introduction to an Asterisque volume, we give a short discussion of the historical developments in the study of nonlinear covering groups, touching on their structure theory, representation theory and the theory of automorphic forms. Th