Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCombinatorics of generalized exponents
167
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.
rate research
Read More
Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers $m_1, m_2, ..., m_k$ which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A_n, B_n, C_n and some other types. When I is zero, we recover the usual exponents of G by Kostant and one of our conjectures reduces to a well-known factorization of the Poincare polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.
We introduce and compute the generalized disconnection exponents $eta_kappa(beta)$ which depend on $kappain(0,4]$ and another real parameter $beta$, extending the Brownian disconnection exponents (corresponding to $kappa=8/3$) computed by Lawler, Schramm and Werner 2001 (conjectured by Duplantier and Kwon 1988). For $kappain(8/3,4]$, the generalized disconnection exponents have a physical interpretation in terms of planar Brownian loop-soups with intensity $cin (0,1]$, which allows us to obtain the first prediction of the dimension of multiple points on the cluster boundaries of these loop-soups. In particular, according to our prediction, the dimension of double points on the cluster boundaries is strictly positive for $cin(0,1)$ and equal to zero for the critical intensity $c=1$, leading to an interesting open question of whether such points exist for the critical loop-soup. Our definition of the exponents is based on a certain general version of radial restriction measures that we construct and study. As an important tool, we introduce a new family of radial SLEs depending on $kappa$ and two additional parameters $mu, u$, that we call radial hypergeometric SLEs. This is a natural but substantial extension of the family of radial SLE$_kappa(rho)s$.
An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables.
We solve the normal ordering problem for (A* A)^n where A* (resp. A) are one mode deformed bosonic creation (resp. annihilation) operators satisfying [A,A*]=[N+1]-[N]. The solution generalizes results known for canonical and q-bosons. It involves combinatorial polynomials in the number operator N for which the generating functions and explicit expressions are found. Simple deformations provide examples of the method.
We examine partition zeta functions analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahons partial fraction decomposition of the generating function for partitions of fixed length.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
