Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #𝖯. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. This includes the result that plethysm coefficients are in #BQP, which was only known in certain cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, thus unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform; recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures the approaches of our and previous work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

Abstract Littlewood-Richardson, Kronecker and plethysm coefficients are fundamental multiplicities of interest in Representation Theory and Algebraic Combinatorics. Determining a combinatorial interpretation for the Kronecker and plethysm coefficients is a major open problem and prompts the consideration of their computational complexity. Recently, it was shown that they behave relatively well with respect to quantum computation, and for some large families there are polynomial-time quantum algorithms (Larocca and Havlicek in Quantum algorithms for representation-theoretic multiplicities, 2024. arXiv:2407.17649 ) (also Bravyi et al. in PRX Quantum 5(1):010329, 2024). In this paper, we show that for many of those cases the Kronecker and plethysm coefficients can also be computed in polynomial time via classical algorithms, thereby refuting some of the conjectures in Larocca and Havlicek (2024). This vastly limits the cases in which the desired superpolynomial quantum speedup could be achieved.

We prove that deciding the vanishing of the character of the symmetric group is C=P-complete. We use this hardness result to prove that the absolute value and also the square of the character are not contained in #P, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof we conclude that deciding positivity of the character is PP-complete under many-one reductions, and hence PH-hard under Turing-reductions.

Abstract. We explore inequalities on linear extensions of posets and make them effective in different ways. First, we study the Björner–Wachs inequality and generalize it to inequalities on order polynomials and their q-analogues via direct injections and Fortuin–Kasteleyn–Ginibre inequalities. Second, we give an injective proof of Sidorenko’s inequality with computational complexity significance, namely, that the difference is in #P. Third, we generalize actions of Coxeter groups on restricted linear extensions, leading to vanishing and uniqueness conditions for the generalized Stanley inequality. We also establish several new inequalities on order polynomials and prove an asymptotic version of Graham’s inequality.

We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics throughout the regime $\ell = \Theta (m)$ and $n = \Theta (m^2)$, while previously sharp asymptotics were derived by Takács [ J. Statist. Plann. Inference, 14 (1986), pp. 123--142] only in the regime where $|n - \ell m /2| = O(\sqrt{\ell m (\ell + m)})$ using a local central limit theorem. Our approach is to solve a related large deviation problem: we describe the tilted measure that produces configurations whose bounding rectangle has the given aspect ratio and is filled to the given proportion. Our results are sufficiently sharp to yield the first asymptotic estimates on the consecutive differences of these numbers when $n$ is increased by one and $m, \ell$ remain the same, hence significantly refining Sylvester's unimodality theorem and giving effective asymptotic estimates for related Kronecker and plethysm coefficients from representation theory.

The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In [A. H. Morales, I. Pak, and G. Panova, Hook Formulas for Skew Shapes I. $q$-Analogues and Bijections] we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations.