Schur log-concavity and the quantum Pascal triangle

2025/09/26 by Gutiérrez, Álvaro, Krattenthaler, Christian
#Combinatorics (math.CO) #FOS: Mathematics #Primary 05E05 #Secondary 05A30

paper · doi:10.48550/arxiv.2509.22648

Abstract

We say a sequence f₀, f₁, f₂, … of symmetric functions is Schur log-concave if f_n² - f_n-1f_n+1 is Schur positive for all n≥1. We conjecture that a very general class of sequences of Schur functions satisfies this property, and show it for sequences of Schur functions indexed by partitions with growing first part and column. Our findings are related to work of Lam, Postnikov and Pylyavskyy on Schur positivity, and of Butler, Sagan, and the second author on q-log-concavity.

Citations

• Plethysms of symmetric functions and representations of SL₂(ℂ)
• Schur-positivity in a Square
• On products of sl_n characters and support containment
• Schur positivity and Schur log-concavity
• Some positive differences of products of Schur functions
• Eigenvalues, singular values, and Littlewood-Richardson coefficients

Related

• Schur positivity and Schur log-concavity
• Logarithmic concavity of Schur and related polynomials
• Schur positivity and log-concavity related to longest increasing\n subsequences
• Schur Positivity and the q-Log-convexity of the Narayana Polynomials
• k-Schur expansions of Catalan functions