30 research outputs found
A new characterization of the exceptional Lie algebras
For a simple Lie algebra, over , we consider the weight which is
the sum of all simple roots and denote it . We formally use
Kostant's weight multiplicity formula to compute the "dimension" of the
zero-weight space. In type , is the highest root, and
therefore this dimension is the rank of the Lie algebra. In type , this is
the defining representation, with dimension equal to 1. In the remaining cases,
the weight is not dominant and is not the highest weight of an
irreducible finite-dimensional representation. Kostant's weight multiplicity
formula, in these cases, is assigning a value to a virtual representation. The
point, however, is that this number is nonzero if and only if the Lie algebra
is classical. This gives rise to a new characterization of the exceptional Lie
algebras as the only Lie algebras for which this value is zero.Comment: 22 pages, 2 figures, and 8 table
On (t,r) Broadcast Domination Numbers of Grids
The domination number of a graph is the minimum cardinality of
any subset such that every vertex in is in or adjacent to
an element of . Finding the domination numbers of by grids was an
open problem for nearly 30 years and was finally solved in 2011 by Goncalves,
Pinlou, Rao, and Thomass\'e. Many variants of domination number on graphs have
been defined and studied, but exact values have not yet been obtained for
grids. We will define a family of domination theories parameterized by pairs of
positive integers where which generalize domination
and distance domination theories for graphs. We call these domination numbers
the broadcast domination numbers. We give the exact values of
broadcast domination numbers for small grids, and we identify upper bounds for
the broadcast domination numbers for large grids and conjecture that
these bounds are tight for sufficiently large grids.Comment: 28 pages, 43 figure
On singularity and normality of regular nilpotent Hessenberg varieties
Regular nilpotent Hessenberg varieties form an important family of
subvarieties of the flag variety, which are often singular and sometimes not
normal varieties. Like Schubert varieties, they contain distinguished points
called permutation flags. In this paper, we give a combinatorial
characterization for a permutation flag of a regular nilpotent Hessenberg
variety to be a singular point. We also apply this result to characterize
regular nilpotent Hessenberg varieties which are normal algebraic varieties.Comment: 36 page
A Formula for the Cohomology and K-Class of a Regular Hessenberg Variety
Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on h than here