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

Breaking down the reduced Kronecker coefficients

We resolve three interrelated problems on reduced Kronecker coefficients $\overline{g}(\alpha ,\beta ,\gamma )$. First, we disprove the 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 ${\textrm{\#P}}$-hard, i.e. ${\textrm{\#P}}$-hard when the input $(\lambda ,\mu ,\nu )$ is in unary

Random discrete concave functions on an equilateral lattice with periodic Hessians

Motivated by connections to random matrices, Littlewood-Richardson coefficients and tilings, we study random discrete concave functions on an equilateral lattice. We show that such functions having a periodic Hessian of a fixed average value $- s$ concentrate around a quadratic function. We consider the set of all concave functions $g$ on an equilateral lattice $\mathbb L$ that when shifted by an element of $n \mathbb L$ have a periodic discrete Hessian, with period $n \mathbb L$. We add a convex quadratic of Hessian $s$; the sum is then periodic with period $n \mathbb L$, and view this as a mean zero function $g$ on the set of vertices $V(\mathbb{T}_n)$ of a torus $\mathbb{T}_n := \frac{\mathbb{Z}}{n\mathbb{Z}}\times \frac{\mathbb{Z}}{n\mathbb{Z}}$ whose Hessian is dominated by $s$. The resulting set of semiconcave functions forms a convex polytope $P_n(s)$. The $\ell_\infty$ diameter of $P_n(s)$ is bounded below by $c(s) n^2$, where $c(s)$ is a positive constant depending only on $s$. Our main result is that under certain conditions, that are met for example when $s_0 = s_1 \leq s_2$, for any $\epsilon > 0,$ we have $$\lim_{n \rightarrow 0} \mathbb{P}\left[\|g\|_\infty > n^{\frac{7}{4} + \epsilon}\right] = 0$$ if $g$ is sampled from the uniform measure on $P_n(s)$. Each $g \in P_n(s)$ corresponds to a kind of honeycomb. We obtain concentration results for these as well.Comment: 56 pages. arXiv admin note: substantial text overlap with arXiv:1909.0858

Gauged fermionic matrix quantum mechanics

We consider the gauged free fermionic matrix model, for a single fermionic matrix. In the large $N$ limit this system describes a $c=1/2$ chiral fermion in $1+1$ dimensions. The Gauss' law constraint implies that to obtain a physical state, indices of the fermionic matrices must be fully contracted, to form a singlet. There are two ways in which this can be achieved: one can consider a trace basis formed from products of traces of fermionic matrices or one can consider a Schur function basis, labeled by Young diagrams. The Schur polynomials for the fermions involve a twisted character, as a consequence of Fermi statistics. The main result of this paper is a proof that the trace and Schur bases coincide up to a simple normalization coefficient that we have computed.Comment: 20 page

Backgrounds from Tensor Models: A Proposal

Although tensor models are serious candidates for a theory of quantum gravity, a connection with classical spacetimes have been elusive so far. This paper aims to fill this gap by proposing a neat connection between tensor theory and Euclidean gravity at the classical level. The main departure from the usual approach is the use of Schur invariants (instead of monomial invariants) as manifold partners. Classical spacetime features can be identified naturally on the tensor side in this new setup. A notion of locality is shown to emerge through Ward identities, where proximity between spacetime points translates into vicinity between Young diagram corners.Comment: 33 page