On the complexity of computing Kronecker coefficients

We study the complexity of computing Kronecker coefficients $g(\lambda,\mu,\nu)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three partitions. When $M = O(1)$, i.e. one of the partitions is hook-like, the bounds are linear in $\log N$, but depend exponentially on $\ell$. Moreover, similar bounds hold even when $M=e^{O(\ell)}$. By a separate argument, we show that the positivity of Kronecker coefficients can be decided in $O(\log N)$ time for a bounded number $\ell$ of parts and without restriction on $M$. Related problems of computing Kronecker coefficients when one partition is a hook, and computing characters of $S_n$ are also considered.Comment: v3: incorporated referee's comments; accepted to Computational Complexit

The Saxl Conjecture and the Dominance Order

In 2012 Jan Saxl conjectured that all irreducible representations of the symmetric group occur in the decomposition of the tensor square of the irreducible representation corresponding to the staircase partition. We make progress on this conjecture by proving the occurrence of all those irreducibles which correspond to partitions that are comparable to the staircase partition in the dominance order. Moreover, we use our result to show the occurrence of all irreducibles corresponding to hook partitions. This generalizes results by Pak, Panova, and Vallejo from 2013.Comment: 11 page

Complexity of short Presburger arithmetic

We study complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integers involved in the inequalities. We prove that assuming Kannan's partition can be found in polynomial time, the satisfiability of Short-PA sentences can be decided in polynomial time. Furthermore, under the same assumption, we show that the numbers of satisfying assignments of short Presburger sentences can also be computed in polynomial time

Breaking down the reduced Kronecker coefficients

We resolve three interrelated problems on \emph{reduced Kronecker coefficients} $\overline{g}(\alpha,\beta,\gamma)$. First, we disprove the \emph{saturation property} which states that $\overline{g}(N\alpha,N\beta,N\gamma)>0$ implies $\overline{g}(\alpha,\beta,\gamma)>0$ for all $N>1$. Second, we esimate the maximal $\overline{g}(\alpha,\beta,\gamma)$, over all $|\alpha|+|\beta|+|\gamma| = n$. Finally, we show that computing $\overline{g}(\lambda,\mu,\nu)$ is strongly $\# P$-hard, i.e. $\#P$-hard when the input $(\lambda,\mu,\nu)$ is in unary.Comment: 5 page

Representations of the symmetric group are decomposable in polynomial time

We introduce an algorithm to decompose orthogonal matrix representations of the symmetric group over the reals into irreducible representations, which as a by-product also computes the multiplicities of the irreducible representations. The algorithm applied to a $d$-dimensional representation of $S_n$ is shown to have a complexity of $O(n^2 d^3)$ operations for determining multiplicities of irreducible representations and a further $O(n d^4)$ operations to fully decompose representations with non-trivial multiplicities. These complexity bounds are pessimistic and in a practical implementation using floating point arithmetic and exploiting sparsity we observe better complexity. We demonstrate this algorithm on the problem of computing multiplicities of two tensor products of irreducible representations (the Kronecker coefficients problem) as well as higher order tensor products. For hook and hook-like irreducible representations the algorithm has polynomial complexity as $n$ increases

Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficients

We give necessary conditions for the positivity of Littlewood-Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if $S^\lambda(V)$ appears as a summand in the decomposition into irreducibles of $S^\mu(S^\nu(V))$, then $\nu$'s diagram is contained in $\lambda$'s diagram.Comment: 11 pages, 7 figure

Plane partitions and the combinatorics of some families of reduced Kronecker coefficients.

International audienceWe compute the generating function of some families of reduced Kronecker coefficients. We give a combi- natorial interpretation for these coefficients in terms of plane partitions. This unexpected relation allows us to check that the saturation hypothesis holds for the reduced Kronecker coefficients of our families. We also compute the quasipolynomial that govern these families, specifying the degree and period. Moving to the setting of Kronecker co- efficients, these results imply some observations related to the rate of growth experienced by the families of Kronecker coefficients associated to the reduced Kronecker coefficients already studied

Bounds on Kronecker and qq-binomial coefficients

We present a lower bound on the Kronecker coefficients for tensor squares of the symmetric group via the characters of~$S_n$, which we apply to obtain various explicit estimates. Notably, we extend Sylvester's unimodality of $q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in~$q$ to derive sharp bounds on the differences of their consecutive coefficients. We then derive effective asymptotic lower bounds for a wider class of Kronecker coefficients.Comment: version May 2016: improved the effective constants. To appear in JCTA. This paper is an extension of parts of the earlier paper "Bounds on the Kronecker coefficients" arXiv:1406.2988, which also contains stability result