268 research outputs found
Grossberg-Karshon twisted cubes and hesitant jumping walk avoidance
Let be a complex simply-laced semisimple algebraic group of rank and
a Borel subgroup. Let be a word and let be a sequence of non-negative integers. Grossberg and
Karshon introduced a virtual lattice polytope associated to and
called a twisted cube, whose lattice points encode the character
of a -representation. More precisely, lattice points in the twisted cube,
counted with sign according to a certain density function, yields the character
of the generalized Demazure module determined by and . In recent work, the author and Harada described precisely when the
Grossberg-Karshon twisted cube is untwisted, i.e., the twisted cube is a closed
convex polytope, in the situation when the integer sequence
comes from a weight of . However, not every integer sequence
comes from a weight of . In the present paper, we interpret
untwistedness of Grossberg-Karshon twisted cubes associated to any word
and any integer sequence using the combinatorics of
and . Indeed, we prove that the Grossberg-Karshon
twisted cube is untwisted precisely when is
hesitant-jumping--walk-avoiding.Comment: Keywords: Grossberg-Karshon twisted cubes, pattern avoidance,
character formula, generalized Demazure modules. arXiv admin note: text
overlap with arXiv:1407.854
Algebraic and geometric properties of flag Bott-Samelson varieties and applications to representations
We introduce the notion of flag Bott-Samelson variety as a generalization of
Bott-Samelson variety and flag variety. Using a birational morphism from an
appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute
Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string
polytopes, which are applied to give polyhedral expressions for irreducible
decompositions of tensor products of -modules. Furthermore, we show that
flag Bott-Samelson varieties are degenerated into flag Bott manifolds with
higher rank torus actions, and find the Duistermaat-Heckman measures of the
moment map images of flag Bott-Samelson varieties with the torus action
together with invariant closed -forms
Maximum Gap in (Inverse) Cyclotomic Polynomial
Let denote the maximum of the differences (gaps) between two
consecutive exponents occurring in a polynomial . Let denote the
-th cyclotomic polynomial and let denote the -th inverse
cyclotomic polynomial. In this note, we study and where
is a product of odd primes, say , etc. It is trivial to
determine , and . Hence the
simplest non-trivial cases are and . We
provide an exact expression for We also provide an exact
expression for
under a mild condition. The condition is almost always
satisfied (only finite exceptions for each ). We also provide a lower
bound and an upper bound for
- β¦