Related papers: Quantum Algorithms for Representation-Theoretic Mu…
We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group…
We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our…
We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial. Moreover, we prove the…
We settle the question of where exactly the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a…
The two variable Kostka functions are the scalar products of the Macdonald polynomials with the Schur polynomials with respect to the scalar product which makes the Hall-Littlewood polynomials pairwise orthogonal. A conjecture of Macdonald…
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same…
In two papers, B\"urgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix…
Bi-partite ribbon graphs arise in organising the large $N$ expansion of correlators in random matrix models and in the enumeration of observables in random tensor models. There is an algebra $\mathcal{K}(n)$, with basis given by bi-partite…
Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous…
The computation of Kronecker coefficients is a challenging problem with a variety of applications. In this paper we present an approach based on methods from symplectic geometry and residue calculus. We outline a general algorithm for the…
The Kronecker coefficients are the decomposition multiplicities of the tensor product of two irreducible representations of the symmetric group. Unlike the Littlewood--Richardson coefficients, which are the analogues for the general linear…
Given an positive integer $k$, let $n:=\binom{k+1}{2}$. In 2012, during a talk at UCLA, Jan Saxl conjectured that all irreducible representations of the symmetric group $S_n$ occur in the decomposition of the tensor square of the…
I study the class of problems efficiently solvable by a quantum computer, given the ability to "postselect" on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic…
We give formulae for computing Kronecker coefficients occurring in the expansion of $s_{\mu}*s_{\nu}$, where both $\mu$ and $\nu$ are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study…
The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups; namely, given three partitions $\lambda, \mu, \tau$ of $n$, the multiplicity of $\lambda$ in $\mu \otimes \tau$ is…
Analytic combinatorics studies the asymptotic behaviour of sequences through the analytic properties of their generating functions. This article provides effective algorithms required for the study of analytic combinatorics in several…
We present a computational approach to studying the structure of the representation ring of the symmetric group in dimension six. The Kronecker coefficients and all power formulae of irreducible representations of $S_6$ are computed using…
Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by $r$-partitions $\lambda, \mu$ and a sign $+, -$. It is known that Kostka polynomials have…
We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of…
We present classical and quantum algorithms based on spectral methods for a problem in tensor principal component analysis. The quantum algorithm achieves a quartic speedup while using exponentially smaller space than the fastest classical…