A3-Weight multiplicity formula

1975 ◽  
Vol 7 (2) ◽  
pp. 275-279
Author(s):  
T. Palev
Keyword(s):  
Weight Multiplicity ◽  
Multiplicity Formula

scholarly journals When is the $q$-Multiplicity of a Weight a Power of $q$?

10.37236/8758 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Pamela E. Harris ◽  
Margaret Rahmoeller ◽  
Lisa Schneider ◽  
Anthony Simpson
Keyword(s):  
Lie Algebras ◽  
Dynkin Diagram ◽  
Fibonacci Numbers ◽  
Simple Lie Algebras ◽  
Weight Multiplicity ◽  
Highest Weight ◽  
Multiplicity Formula ◽  
Exhaustive List

Berenshtein and Zelevinskii provided an exhaustive list of pairs of weights $(\lambda,\mu)$ of simple Lie algebras $\mathfrak{g}$ (up to Dynkin diagram isomorphism) for which the multiplicity of the weight $\mu$ in the representation of $\mathfrak{g}$ with highest weight $\lambda$ is equal to one. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity for subsets of these pairs of weights and show that, in these cases, the cardinality of these contributing sets is enumerated by (multiples of) Fibonacci numbers. We conclude by using these results to compute the associated $q$-multiplicity for the pairs of weights considered, and conjecture that in all cases the $q$-multiplicity of such pairs of weights is given by a power of $q$.

scholarly journals Weight multiplicity formulas for bivariate representations of classical Lie algebras

2018 ◽  
Vol 59 (8) ◽  
pp. 081705 ◽  
Author(s):  
Emilio A. Lauret ◽  
Fiorela Rossi Bertone
Keyword(s):  
Lie Algebras ◽  
Weight Multiplicity

scholarly journals Bivariate systems of polynomial equations with roots of high multiplicity

2021 ◽  
Author(s):  
I. Nikitin
Keyword(s):  
High Multiplicity ◽  
Polynomial Equations ◽  
Support Sets ◽  
Multiplicity Formula ◽  
System Of Polynomial Equations ◽  
Systems Of Polynomial Equations

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

scholarly journals Computing subschemes of the border basis scheme

2020 ◽  
Vol 30 (08) ◽  
pp. 1671-1716
Author(s):  
Martin Kreuzer ◽  
Le Ngoc Long ◽  
Lorenzo Robbiano
Keyword(s):  
Complete Intersection ◽  
General Problem ◽  
Hilbert Scheme ◽  
Affine Space ◽  
Rational Points ◽  
Open Covering ◽  
Natural Question ◽  
Quadratic Equations ◽  
Multiplicity Formula

A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional [Formula: see text]-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

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