Monomial Bounds and Redundancy in Equivariant Schubert Calculus on Lagrangian Grassmannians

.AMNNAH ABUOJAYLAH ALHADI .. ².NAJIMUDIN AMIR MAWLOUD  ..³. WEDYAN MUKTAR A ELJIAFA      ⁽¹⁺³⁾ Faculty of Education / University of Zawiya. Libya..^{. (2)}Faculty of Science / University of Zawiya, Libya

¹am.rasheed@zu.edu.ly           ²N.Shalghoum@zu.edu.ly            ³ w.eljiafari@zu.edu.ly

https://orcid.org/0009-0001-3676-4673 ..  https://orcid.org/0009-0004-7747-9969 .. https://orcid.org/0009-000-8747-5970

Abstract:

We prove an unconditional upper bound for the number of distinct monomials in any equivariant Schubert coefficient on :

where . The bound is sharp and follows solely from the polynomial nature of the coefficient. We introduce a conjectural edge-labeled tableau model (Conjecture 3.1) for the Lagrangian Grassmannian. Under this conjecture, we obtain a complementary bound  and the refined estimate

We define a redundancy index  and prove, via the pigeonhole principle,

When , the bound forces the maximum redundancy to be at least . We characterize injectivity of the weight map and show that the conjectural model recovers the classical Tamvakis rule [5]. A detailed example for  illustrates the theory.

Keywords: Equivariant Schubert calculus; Lagrangian Grassmannian; edge-labeled tableaux; monomial bounds; strict partitions; redundancy index.