Related papers: A sign-reversing involution for rooted special rim…

Let $k$ be a field, let $R$ be a commutative ring, and assume the exponential characteristic of $k$ is invertible in $R$. In this note, we prove that isomorphisms in Voevodsky's triangulated category of motives $\mathcal{DM}(k;R)$ are…

The central topic is this question: is a given $k$-\'etale algebra $\prod_lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$? Our main tool is a {\it twisting lemma} that reduces the problem to…

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2].…

Let $W$ be a Coxeter group, and for $u,v\in W$, let $R_{u,v}(q)$ be the Kazhdan-Lusztig $R$-polynomial indexed by $u$ and $v$. In this paper, we present a combinatorial proof of the inversion formula on $R$-polynomials due to Kazhdan and…

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ... (3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. Mills, Robbins, and Rumsey also made the stronger…

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the…

Lie algebra involutions and their fixed-point subalgebras give rise to symmetric spaces and real forms of complex Lie algebras, and are well-studied in the context of symmetrizable Kac-Moody algebras. In this paper we study a…

In this paper, we prove certain multiplicity one theorems and define twisted gamma factors for irreducible generic cuspidal representations of split $G_2$ over finite fields $k$ of odd characteristic. Then we prove the first converse…

We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for…

We study two different one-parameter generalizations of Littlewood--Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions…

We revisit Eisenstein's geometric proof of quadratic reciprocity and make explicit the involutive symmetry underlying Eisenstein's lattice-point argument. Building on Gauss's lemma, we interpret the Legendre symbols as counts of lattice…

In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group. This model, which will be referred to as the alcove path model, can be viewed as a discrete…

Computational reflection allows us to turn verified decision procedures into efficient automated reasoning tools in proof assistants. The typical applications of such methodology include mathematical structures that have decidable theory…

This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a…

Following ideas of Bloch, Esnault, and Kerz, the deformational part of Grothendieck's variational Hodge conjecture is established for proper, smooth schemes over $K[[t]]$, where $K$ is an algebraic extension of the rational numbers. The…

We give an introduction to the "stable algebra of matrices" as related to certain problems in symbolic dynamics. We consider this stable algebra (especially, shift equivalence and strong shift equivalence) for matrices over general rings as…

We give an invariant construction of reduced HOMFLY homology for arbitrary links reduced at components of arbitrary color and prove some structural properties relating this invariant to unreduced HOMFLY homology. Combined with previous…

We prove K-theoretic and shifted K-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for pairs of stable Grothendieck polynomials…

We formulate and prove the Remodeling Conjecture with descendants, which is a version of all-genus equivariant descendant mirror symmetry for semi-projective toric Calabi-Yau 3-orbifolds with integral structures. We construct an isomorphism…

Odd index pairings of $K_1$-group elements with Fredholm modules are of relevance in index theory, differential geometry and applications such as to topological insulators. For the concrete setting of operators on a Hilbert space over a…