Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.14609

Constructed tableaux and a new shifted Littlewood-Richardson rule

Santiago Estupiñán-Salamanca·Oliver Pechenik
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a new Littlewood-Richardson rule for the Schubert structure coefficients of isotropic Grassmannians, equivalently for the multiplication of $P$-Schur functions. Serrano (2010) previously gave a formula in terms of classes in his shifted plactic monoid. However, this formula is challenging to use because of the difficulty of characterizing shifted plactic classes. We give the first algebraic proof of this formula. We then use it to obtain a new rule that is easy to implement. Our rule is based on identifying a subtle analogue of Yamanouchi tableaux, which we characterize. We show that for some families of structure coefficients, our rule leads to an algorithm with exponentially better time complexity than the original rule of Stembridge (1989).

Explore related subjects

math.COmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Santiago Estupiñán-Salamanca, Oliver Pechenik. 2025-06-19. Constructed tableaux and a new shifted Littlewood-Richardson rule. https://arxiv.org/abs/2503.14609

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On nut graphs with two vertex and three edge orbits

Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Bašić, Fowler and Pisanski proved that the automorphism group of a nut graph has more orbits on the edge set than on the vertex set. They classified all orders for which a vertex-transitive nut graph with precisely two edge orbits exists, and conjectured that a nut graph with two vertex and three edge orbits exists for each non-prime order $n \ge 9$. Motivated by this conjecture, we introduce a very general construction that provides graphs with the desired symmetry properties, and we determine some sufficient spectral and structural conditions under which they are nut graphs. The construction yields infinite families of examples and confirms the above conjecture for all odd non-prime orders up to $2\,500$ and for at least $99.8$ percent of all odd non-prime orders up to a million. Finally, we present some additional interesting examples of nut graphs with two vertex and three edge orbits that do not arise from this construction.

math.CO↗
arXiv2026

On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $λ(n,d)$ be the minimal number of vertices of a triangulation of the $n$-sphere which admits a degree $d$ simplicial map onto the boundary of the $(n+1)$-simplex. We show that for $h=\lfloor\frac{n+1}2\rfloor$, the function $λ(n,d)^h$ has linear order of growth in $d$, answering a question of O. Musin. All triangulations we obtained are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.

math.CO↗