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We introduce two new partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. Both posets are ranked by the major index statistic offset by a fixed shift.…

It is well-known from the representation theory of particle physics that the tensor product of two fundamental representation of SU(2) and SU(3) group can be decomposed to obtain the desired spectrum of the physical states. In this paper,…

We give a direct combinatorial proof that the product of two descent classes in a symmetric group is a sum of descent classes. The proof is based on the fact that the group product gives a covering map when descent classes are endowed with…

We study Landau-Ginzburg orbifolds $(f,G)$ with $f=x_1^n+\ldots+x_N^n$ and $G=S\ltimes G^d$, where $S\subseteq S_N$ and $G^d$ is either the maximal group of scalar symmetries of $f$ or the intersection of the maximal diagonal symmetries of…

Let $S$ be a smooth projective surface on a smooth threefold $X$ such that $X$ has Picard rank 1 and NS$(S)$ is generated by the restriction of divisors from X. We show that if $X$ satisfies the Bogomolov-Gieseker type inequality for tilt…

We calculate the representation growth zeta function of the discrete Heisenberg group over the integers of a quadratic number field. This is done by forming equivalence classes of representations, called twist iso-classes, and explicitly…

We prove that two n-by-n matrices A and B have their rank-k numerical ranges $\Lambda_k(A)$ and $\Lambda_k(B)$ equal to each other for all k, $1\le k\le \lfloor n/2\rfloor+1$, if and only if their Kippenhahn polynomials…

We prove the existence of infinitely many real and imaginary fields whose 5-rank of the class group is >=3.

We fill the gaps in A. Gica's determination of all the odd positive integers $d$ for which the number of distinct prime divisors of $f_d(x)=d+x^2$ is less than or equal to $2$ for all the positive and odd integers $x\leq\sqrt{d}$. We also…

We prove that there is a unique $R$-equivalence class on every del Pezzo surface of degree $4$ defined over the Laurent field $K=k((t))$ in one variable over an algebraically closed field $k$ of characteristic not equal to $2$ or $5$. We…

Let $k$ be a perfect field of characteristic $\neq 2$. We prove that the Schmidt rank (also known as strength) of a quartic polynomial $f$ over $k$ is bounded above in terms of only the Schmidt rank of $f$ over $\overline{k}$, an algebraic…

We consider indecomposable representations of the Klein four group over a field of characteristic $2$ and of a cyclic group of order $pm$ with $p,m$ coprime over a field of characteristic $p$. For each representation we explicitly describe…

The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are…

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas…

We construct an infinite family of imaginary bicyclic biquadratic number fields $k$ with the 2-ranks of their 2-class groups are $\geq3$, whose strongly ambiguous classes of $k/Q(i)$ capitulate in the absolute genus field $k^{(*)}$, which…

We prove some eigenvalue inequalities for positive semidefinite matrices partitioned into four blocks. The inradius of the numerical range of the off-diagonal block contributes to these estimates. Some related norm inequalities are given…

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree $d$ have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain…

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar…

Moment inequality for quadratic forms of random vectors is of particular interest in covariance matrix testing and estimation problems. In this paper, we prove a Rosenthal-type inequality, which exhibits new features and certain improvement…

To what extent does the maximal subfield spectrum of a division algebra determine the isomorphism class of that algebra? It has been shown that over some fields a quaternion division algebra's isomorphism class is largely if not entirely…