Related papers: Constructive Method for Finding the Coefficients o…

Let D be a masa in B(H) where H is a separable Hilbert space. We find real numbers \eta_0 < \eta_1 < \eta_2 < ... < \eta_6 so that for every bounded, normal D-bimodule map {\Phi} on B(H) either ||\Phi|| > \eta_6, or ||\Phi|| = \eta_k for…

When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial $E_{\lambda}(x;q;0)$ is symmetric and related to a modified Hall--Littlewood polynomial. We show that whenever all parts of the integer partition $\lambda$ is…

In this work we show that given a connectivity graph $G$ of a $[[n,k,d]]$ quantum code, there exists $\{K_i\}_i, K_i \subset G$, such that $\sum_i |K_i|\in \Omega(k), \ |K_i| \in \Omega(d)$, and the $K_i$'s are $\tilde{\Omega}(…

We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ is a composition. These are polynomials which are…

In a previous paper Hua-Jost-Liu, we have applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincar\'e inequality on these…

In \cite{HRW15}, Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $\Delta_{e_k}e_n$. It is called the Delta Conjecture. It was recently proved in \cite{GHRY} that the…

We revisit a double-scaled limit of the superconformal index of ${\cal N}=2$ superconformal field theories (SCFTs) which generalizes the Schur index. The resulting partition function, $\hat {\cal Z}(q,\alpha)$, has a standard $q$-expansion…

In a recent article (arXiv:1507.03499) (joint with Alon Regev) we studied sums of squares of characters Chi(L,M) of the Symmetric Group over shapes L that are two-rowed, and shapes L that are hook shapes, and M is an arbitrary shape that…

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable "diagonal…

Let $\mathfrak l:= \mathfrak q(n)\times\mathfrak q(n)$, where $\mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $\mathfrak q(n)$, and hence is equipped with a…

This paper deals with the $\lambda$-labeling and $L(2,1)$-coloring of simple graphs. A $\lambda$-labeling of a graph $G$ is any labeling of the vertices of $G$ with different labels such that any two adjacent vertices receive labels which…

Two methods of constructing 2D Toda $\tau$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group…

One of the central open problems in both algebraic combinatorics and representation theory is to find a positive combinatorial rule for Kronecker coefficients $ g_{\lambda \, \mu \, \nu}$. A notable advance in this direction is due to…

Given a skew diagram $\gamma/\lambda$, we determine a set of lower and upper bounds that a partition $\mu$ must satisfy for Littlewood-Richards coefficients $c^{\gamma}_{\lambda,\mu}>0$. Our algorithm depends on the characterization of…

We examine the interplay of symmetry and topological order in $2+1$ dimensional topological phases of matter. We present a definition of the \it topological symmetry \rm group, which characterizes the symmetry of the emergent topological…

For non-negative integers $n$ and $k$ with $n \ge k$, a {\em $k$-minor} of a partition $\lambda = [\lambda_1, \lambda_2, \dots]$ of $n$ is a partition $\mu = [\mu_1, \mu_2, \dots]$ of $n-k$ such that $\mu_i \le \lambda_i$ for all $i$. The…

The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the…

We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian…

A {\em pointed partition} of $n$ is a pair $(\lambda, v)$ where $\lambda\vdash n$ and $v$ is a cell in its Ferrers diagram. We construct an involution on pointed partitions of $n$ exchanging "hook length" and "part length". This gives a…

Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $\Delta$-bounded if the diameter of each cluster is bounded by $\Delta$. A distribution over $\Delta$-bounded partitions is a $\beta$-padded decomposition if every ball of radius…