相关论文: Fractal behavior of tensor powers of the two dimen…
Given a group $G$ and $V$ a representation of $G$, denote the number of indecomposable summands of $V^{\otimes k}$ by $b_k^{G, V}$. Given a tilting representation $T$ of $\text{SL}_2(K)$ where $K=\overline{K}$ and of characteristic $p>2$,…
We study the decomposition of tensor powers of two dimensional irreducible representations of quantum $\mathfrak{sl}_2$ at even roots of unity into direct sums of tilting modules. We derive a combinatorial formula for multiplicity of…
In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.
We study tensor powers of representations of finite monoids, focusing on the growth behavior of their composition length and the number of indecomposable summands. Special attention is given to diagram monoids such as the Temperley-Lieb,…
We use tilting modules to study the structure of the tensor product of two simple modules for the algebraic group $\SL_2$, in positive characteristic, obtaining a twisted tensor product theorem for its indecomposable direct summands.…
Tensor products of quiver representations have been extensively studied; typical examples include the pointwise tensor product and the tensor product induced by the coalgebra structure of path algebras. In this paper, we investigate the…
We give explicit formulas for the asymptotic growth rate of the number of summands in tensor powers in certain monoidal categories with finitely many indecomposable objects, and related structures.
We obtain the decomposition of the tensor space $\mathfrak{sl}_n^{\otimes k}$ as a module for $\mathfrak{sl}_n$, find an explicit formula for the multiplicities of its irreducible summands, and (when $n \ge 2k$) describe the centralizer…
In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the result of paper we get the known properties…
Let $K$ be an algebraically closed field of characteristic $2$, $G$ be the algebraic group $\mathrm{SL}_2$ over $K$, and $V$ be the natural representation of $G$. Let $b_k^{G,V}$ denote the number of $G$-indecomposable factors of…
In this article, we have employed fractal formalism to calculate the Fracture Functions of the Leading neutron produced in \textit{ep} collisions. The fractal concept describes the self-similar behavior of the proton structure at Leading…
We give a general asymptotic formula for the growth rate of the number of indecomposable summands in the tensor powers of representations of finite groups, over a field of arbitrary characteristic. In characteristic zero we obtain…
We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric…
In this paper we introduce a novel family of Markov chains on the simple representations of $\mathrm{SL}_2(\mathbb{F}_p)$ in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable…
In a categorification of tensor products of fundamental representations of quantum sl(k) via highest weight categories, the indecomposable tilting modules descend to the canonical basis. Since projective functors map tilting modules to…
Let $M$ be a finite dimensional modular representation of a finite group $G$. We consider the generating function for the non-projective part of the tensor powers of $M$, and we write $\gamma_G(M)$ for the reciprocal of the radius of…
Dave Benson conjectured in 2020 that if $G$ is a finite $2$-group and $V$ is an odd-dimensional indecomposable representation of $G$ over an algebraically closed field $\Bbbk$ of characteristic $2$, then the only odd-dimensional…
The term fractal describes a class of complex structures exhibiting self-similarity across different scales. Fractal patterns can be created by using various techniques such as finite subdivision rules and iterated function systems. In this…
For every absolutely irreducible orthogonal representation of a twisted form of SL2 over a field of characteristic zero, we compute the "unique" symmetric bilinear form that is invariant under the group action. We also prove the analogous…
Small-angle scattering (SAS) data which show a succession of power-law decays with decreasing values of scattering exponents, can be described in terms of fractal structures with positive Lebesgue measure (fat fractals). In this work we…