94 research outputs found
Diagram calculus for a type affine Temperley--Lieb algebra, I
In this paper, we present an infinite dimensional associative diagram algebra
that satisfies the relations of the generalized Temperley--Lieb algebra having
a basis indexed by the fully commutative elements (in the sense of Stembridge)
of the Coxeter group of type affine . Moreover, we provide an explicit
description of a basis for the diagram algebra. In the sequel to this paper, we
show that this diagrammatic representation is faithful. The results of this
paper and its sequel will be used to construct a Jones-type trace on the Hecke
algebra of type affine , allowing us to non-recursively compute leading
coefficients of certain Kazhdan--Lusztig polynomials.Comment: Title and content updated to reflect published version. References
and contact information updated. 28 pages, 26 figure
Impartial achievement and avoidance games for generating finite groups
We study two impartial games introduced by Anderson and Harary and further
developed by Barnes. Both games are played by two players who alternately
select previously unselected elements of a finite group. The first player who
builds a generating set from the jointly selected elements wins the first game.
The first player who cannot select an element without building a generating set
loses the second game. After the development of some general results, we
determine the nim-numbers of these games for abelian and dihedral groups. We
also present some conjectures based on computer calculations. Our main
computational and theoretical tool is the structure diagram of a game, which is
a type of identification digraph of the game digraph that is compatible with
the nim-numbers of the positions. Structure diagrams also provide simple yet
intuitive visualizations of these games that capture the complexity of the
positions.Comment: 28 pages, 44 figures. Revised in response to comments from refere
Impartial avoidance games for generating finite groups
We study an impartial avoidance game introduced by Anderson and Harary. The
game is played by two players who alternately select previously unselected
elements of a finite group. The first player who cannot select an element
without making the set of jointly-selected elements into a generating set for
the group loses the game. We develop criteria on the maximal subgroups that
determine the nim-numbers of these games and use our criteria to study our game
for several families of groups, including nilpotent, sporadic, and symmetric
groups.Comment: 14 pages, 4 figures. Revised in response to comments from refere
Impartial avoidance and achievement games for generating symmetric and alternating groups
We study two impartial games introduced by Anderson and Harary. Both games
are played by two players who alternately select previously-unselected elements
of a finite group. The first player who builds a generating set from the
jointly-selected elements wins the first game. The first player who cannot
select an element without building a generating set loses the second game. We
determine the nim-numbers, and therefore the outcomes, of these games for
symmetric and alternating groups.Comment: 12 pages. 2 tables/figures. This work was conducted during the third
author's visit to DIMACS partially enabled through support from the National
Science Foundation under grant number #CCF-1445755. Revised in response to
comments from refere
Impartial achievement games for generating nilpotent groups
We study an impartial game introduced by Anderson and Harary. The game is
played by two players who alternately choose previously-unselected elements of
a finite group. The first player who builds a generating set from the
jointly-selected elements wins. We determine the nim-numbers of this game for
finite groups of the form , where is a -group and is a
group of odd order. This includes all nilpotent and hence abelian groups.Comment: 10 pages, 2 figure
The spectrum of nim-values for achievement games for generating finite groups
We study an impartial achievement game introduced by Anderson and Harary. The
game is played by two players who alternately select previously unselected
elements of a finite group. The game ends when the jointly selected elements
generate the group. The last player able to make a move is the winner of the
game. We prove that the spectrum of nim-values of these games is
. This positively answers two conjectures from a previous paper
by the last two authors.Comment: 11 pages, 5 figures, 2 table
- …