Related papers: Multiplicities and tensor product coefficients for…
Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. In a continuation of a previous paper we prove that, if $D=1$ or $D$ is a prime number, the…
In this paper we present Affine.m - program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. Algorithms are based upon the properties of weights and Weyl symmetry.…
This article introduces an algebra of functions in one variable $c$ defined by iterated integrals of two specific differential forms depending on $c$, where the product is the shuffle product. This algebra can be seen as a common…
For each valued quiver $Q$ of Dynkin type, we construct a valued ice quiver $\Delta_Q^2$. Let $G$ be a simple connected Lie group with Dynkin diagram the underlying valued graph of $Q$. The upper cluster algebra of $\Delta_Q^2$ is graded by…
In this paper, we present fast algorithms for the product of two multivariate polynomials in sparse representation. The bit complexity of our algorithms are studied in detail for various types of coefficients, and we derive new complexity…
We study the generating functions of cylindric partitions having profile $c=(c_1, c_2, \ldots, c_r)$ with rank $2$ and levels $2, 3$ and $4$. As a result, we give expressions alternative to Borodin's formula for these generating functions.…
The Weyl modules in the sense of V.Chari and A.Pressley [CP] over the current Lie algebra on an affine variety are studied. We show that local Weyl modules are finite-dimensional and generalize the tensor product decomposition theorem from…
Permutation equivariant neural networks are often constructed using tensor powers of $\mathbb{R}^{n}$ as their layer spaces. We show that all of the weight matrices that appear in these neural networks can be obtained from Schur-Weyl…
This paper develops fast and efficient algorithms for computing Tucker decomposition with a given multilinear rank. By combining random projection and the power scheme, we propose two efficient randomized versions for the truncated…
We complete our theory of weighted $L^p(w_1) \times L^q(w_2) \to L^r(w_1^{r/p} w_2^{r/q})$ estimates for bilinear bi-parameter Calder\'on--Zygmund operators under the assumption that $w_1 \in A_p$ and $w_2 \in A_q$ are bi-parameter weights.…
A systematic method for constructing trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra has been developed by the Brisbane group and…
Let G=GL_n be the general linear group over an algebraically closed field k and let g=gl_n be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[g] be the algebra of regular functions on…
In this paper, the authors establish some weighted estimates for the Calder\'on commutator defined by \begin{eqnarray*} &&\mathcal{C}_{m+1,\,A}(a_1,\dots,a_{m};f)(x) &&\quad={\rm…
Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula…
We extend the maximal unitarity method to amplitude contributions whose cuts define multidimensional algebraic varieties. The technique is valid to all orders and is explicitly demonstrated at three loops in gauge theories with any number…
Let $V$ be an $n$ dimensional vector space over an algebraic closure of a finite field $F_q$ and put $G = GL(V)$. For a positive integer $r$, we consider the variety $X_{uni} = G_{uni} \times V^{r-1}$, on which $G$ acts diagonally.…
Tensor decompositions have become essential tools for feature extraction and compression of multiway data. Recent advances in tensor operators have enabled desirable properties of standard matrix algebra to be retained for multilinear…
In this paper we study general highest weight modules $\mathbb{V}^\lambda$ over a complex finite-dimensional semisimple Lie algebra $\mathfrak{g}$. We present three formulas for the set of weights of a large family of modules…
We study the complexity of computing Kronecker coefficients $g(\lambda,\mu,\nu)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three…
In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension $d$ by means of tensor product formulas. This is useful, for example, in the context of…