Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In t...

In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associ...

In this thesis, I present an associative diagram algebra that is a faithful representation of a particular Temperley–Lieb algebra of type affine C, which has a basis indexed by the fully commutative elements of the Coxeter group of the same type. The Coxeter group of type affine C contains an infinite number of fully commutative elements, and so the corresponding Temperley–Lieb algebra is of in...

In this note we prove that every finite collection of connected algebraic subgroups the group triangular automorphisms affine space generates a solvable subgroup.

We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is...

With any connected affine algebraic group G over an algebraically closed field K of characteristic zero, we associate another connected affine algebraic group D over K and a finite central subgroup F of D such that, up to isomorphism of algebraic groups, affine algebraic groups over K abstractly isomorphic to G are precisely of the form D/α(F )×Ks +, where α is an abstract automorphism of D and...

the subgroups of symplectic groups which fix a non-zero vector of the underlying symplectic space are called affine subgroups., the split extension group $a(4)cong 2^7{:}sp_6(2)$ is the affine subgroup of the symplectic group $sp_8(2)$ of index $255$‎. ‎in this paper‎, ‎we use the technique of the fischer-clifford matrices to construct the character table of the inertia group $2^7{:}o^{-}_{6}(2...

We consider the structure of groups that act on a p−regular tree in a vertex transitive way with the local action (i.e. the action of the vertex stabilizer on the link) isomorphic to the group of affine transformations on a finite affine line. Mathematics Subject Classification (1991): Primary 20E; Secondary 05C25,22E40,20F32

A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements with the coordinates of the ambient space. It is well known that every affine-invariant code of length pm, with p prime, can be realized as an ideal of the ...