Related papers: On rotated Schur-positive sets

In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive…

Recently is has been proved that if $\sigma\in GL_n(R)$ where $R$ is an commutative ring and $n\geq 3$, then each of the elementary transvections $t_{kl}(\sigma_{ij})~(i\neq j,k\neq l)$ is a product of eight $E_n(R)$-conjugates of $\sigma$…

In 2022, Huh-Matherne-M\'esz\'aros-St. Dizier showed that normalized Schur polynomials are Lorentzian, thereby yielding their continuous (resp. discrete) log-concavity on the positive orthant (resp. on their support, in type $A$ root…

We prove new general results on sumsets of sets having Szemer\'edi--Trotter type. This family includes convex sets, sets with small multiplicative doubling, images of sets under convex/concave maps and others.

We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up…

Motivated by the notion of nice graphs, we introduce the concept of strongly nice property, which can be used to study the Schur positivity of symmetric functions. We show that a graph and all its induced subgraphs are strongly nice if and…

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…

Renormalization group theory allows continuous variation of critical exponents along a marginal direction (when there is one), keeping the scaling relations invariant. We propose a super universality hypothesis (SUH) suggesting that, up to…

H.Choi-D.Kim-S.J.Lee and S.J.Lee introduced a new kind of tableaux, semistandard oscillating tableaux (SSOT), around 2024 in the context of Lusztig $q$-weight multiplicities, KR crystals and King tableaux. In this paper, we study generating…

Canon permutations are permutations of the multiset having $k$ copies of each integer between $1$ and $n$, with the property that the subsequences obtained by taking the $j$th copy of each entry, for each fixed $j$, are all the same. For…

The study of bipartite maps (or Grothendieck's dessins d'enfants) is closely connected with geometry, mathematical physics and free probability. Here we study these objects from their permutation factorization formulation using a novel…

Inspired by the classic apolarity theory of symmetric tensors, the aim of this paper is to introduce the Schur apolarity theory, i.e. an apolarity for any irreducible representation of the special linear group $SL(V)$. This allows to…

We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur…

We study a general class of weighted shifts whose weights $\alpha$ are given by $\alpha_n = \sqrt{\frac{p^n + N}{p^n + D}}$, where $p > 1$ and $N$ and $D$ are parameters so that $(N,D) \in (-1, 1)\times (-1, 1)$. Some few examples of these…

It is known from the work of Baik, Deift, and Johansson [1999] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the…

In this paper, we refine a result of Andrews and Merca on truncated pentagonal number series. Subsequently, we establish some positivity results involving Andrews--Gordon--Bressoud identities and $d$-regular partitions. In particular, we…

The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in…

$n$-cycle permutations with small $n$ have the advantage that their compositional inverses are efficient in terms of implementation. They can be also used in constructing Bent functions and designing codes. Since the AGW Criterion was…

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…

In this thesis we introduce a new type of card shuffle called the one-sided transposition shuffle. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a…