268 research outputs found

### Grossberg-Karshon twisted cubes and hesitant jumping walk avoidance

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and
$B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\mathbf \ell
= (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and
Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and
$\mathbf \ell$ called a twisted cube, whose lattice points encode the character
of a $B$-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 $\mathbf i$ and $\mathbf
\ell$. 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 $\mathbf \ell$
comes from a weight $\lambda$ of $G$. However, not every integer sequence
$\mathbf \ell$ comes from a weight of $G$. In the present paper, we interpret
untwistedness of Grossberg-Karshon twisted cubes associated to any word
$\mathbf i$ and any integer sequence $\mathbf \ell$ using the combinatorics of
$\mathbf i$ and $\mathbf \ell$. Indeed, we prove that the Grossberg-Karshon
twisted cube is untwisted precisely when $\mathbf i$ is
hesitant-jumping-$\mathbf \ell$-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 $G$-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 $2$-forms

### Maximum Gap in (Inverse) Cyclotomic Polynomial

Let $g(f)$ denote the maximum of the differences (gaps) between two
consecutive exponents occurring in a polynomial $f$. Let $\Phi_n$ denote the
$n$-th cyclotomic polynomial and let $\Psi_n$ denote the $n$-th inverse
cyclotomic polynomial. In this note, we study $g(\Phi_n)$ and $g(\Psi_n)$ where
$n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to
determine $g(\Phi_{p_1})$, $g(\Psi_{p_1})$ and $g(\Psi_{p_1p_2})$. Hence the
simplest non-trivial cases are $g(\Phi_{p_1p_2})$ and $g(\Psi_{p_1p_2p_3})$. We
provide an exact expression for $g(\Phi_{p_1p_2}).$ We also provide an exact
expression for
$g(\Psi_{p_1p_2p_3})$ under a mild condition. The condition is almost always
satisfied (only finite exceptions for each $p_1$). We also provide a lower
bound and an upper bound for $g(\Psi_{p_1p_2p_3})$

- β¦