arXiv subjects

Rafael Stekolshchik

Publications and source records attributed to Rafael Stekolshchik.

15 recordsLinked to original sources

arXiv2025

Integer quadratic forms and extensions of subsets of linearly independent roots

We consider subsets of linearly independent roots in a certain root system $\varPhi$. Let $S'$ be such a subset, and let $S'$ be associated with any Carter diagram $Γ'$. The main question of the paper: what root $γ\in \varPhi$ can be added to $S'$ so that $S' \cup γ$ is also a subset of linearly independent roots? This extra root $γ$ is called the linkage root. The vector $γ^{\nabla}$ of inner products $\{(γ,τ'_i)\mid τ'_i \in S'\}$ is called the linkage label vector. Let $B_{Γ'}$ be the Cartan matrix associated with $Γ'$. It is shown that $γ$ is a linkage root if and only if $\mathscr{B}^{\vee}_{Γ'}(γ^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_{Γ'}$ is a quadratic form with the matrix inverse to $B_{Γ'}$. The set of all linkage roots for $Γ'$ is called a linkage system and is denoted by $\mathscr{L}(Γ')$. The Cartan matrix associated with any Carter diagram $Γ'$ is conjugate to the Cartan matrix associated with some Dynkin diagram $Γ$, [St23]. The sizes of $\mathscr{L}(Γ')$ and $\mathscr{L}(Γ)$ are the same. Let $W^{\vee}$ be the Weyl group of the quadratic form $\mathscr{B}^{\vee}_{Γ'}$. This group acts on the linkage system and forms several orbits. The sizes and structure of orbits for linkage systems $\mathscr{L}(D_l)$ and $\mathscr{L}(D_l(a_k))$ are presented.

math.RT↗

arXiv2024

Extraspecial pairs in the multiply-laced root systems and calculating structure constants

The notions of special and extraspecial pairs of roots were introduced by Carter for calculating structure constants, [Ca72]. Let $\{r, s\}$ be a special pair of roots for which the structure constant $N(r,s)$ is sought, and let $\{r_1, s_1\}$ be the extraspecial pair of roots corresponding to $\{r, s\}$. Consider the ordered set $\{r_1, r, s, s_1\}$, we will call such a set a quartet. By studying the different quartets, we gain additional insight into the internal structure of the root system. It is shown that for the case $B_n$ we can avoid finding $6$ squares of lengths in the formula for calculating the structure constants. The calculation formula for $B_n$ coincides with the formula for the simply-laced case. For the case $C_n$, it is possible to avoid the calculation of $4$ squares of lengths. The calculation formula for $C_n$ differs from simply-laced case by some parameter, which is fixed for all pairs $\{r, s\}$ with given extraspecial pair $\{r_1, s_1\}$.

math.RT↗

arXiv2023

Decomposition of the longest element of the Weyl group using factors corresponding to the highest roots

Let $\varPhi$ be a root system of a finite Weyl group $W$ with simple roots $Δ$ and corresponding simple reflections $S$. For $J \subseteq S$, denote by $W_J$ the standard parabolic subgroup of $W$ generated by $J$, and by $Δ_J \subseteq Δ$ the subset corresponding to $J$. We show that the longest element of $W$ is decomposed into a product of several ($\le |Δ|$) reflections corresponding to mutually orthogonal roots, each of which is either the highest root of some subset $Δ_J \subseteq Δ$ or is a simple root. For each type of the root system, the factors of the specified decomposition are listed. The relationship between the longest elements of different types is found out. The uniqueness of the considered decomposition is shown. It turns out that subsets of highest roots, which give the decomposition of longest elements in the Weyl group, coincide with the cascade of orthogonal roots constructed by B.Kostant and A.Joseph for calculations in the universal enveloping algebra.

math.RT↗

arXiv2023

Transitions between root subsets associated with Carter diagrams

For any two root subsets associated with two Carter diagrams that have the same $ADE$ type and the same size, we construct the transition matrix that maps one subset to the other. The transition between these two subsets is carried out in some canonical way affecting exactly one root, so that this root is mapped to the minimal element in some root subsystem. The constructed transitions are involutions. It is shown that all root subsets associated with the given Carter diagram are conjugate under the action of the Weyl group. A numerical relationship is observed between enhanced Dynkin diagrams $Δ(E_6)$, $Δ(E_7)$ and $Δ(E_8)$ (introduced by Dynkin-Minchenko) and Carter diagrams. This relationship echoes the $2-4-8$ assertions obtained by Ringel, Rosenfeld and Baez in completely different contexts regarding the Dynkin diagrams $E_6$, $E_7$, $E_8$.

math.RT↗

arXiv2022

Extending Snow's algorithm for computations in the finite Weyl groups

In 1990, D.Snow proposed an effective algorithm for computing the orbits of finite Weyl groups. Snow's algorithm is designed for computation of weights, $W$-orbits and elements of the Weyl group. An extension of Snow's algorithm is proposed, which allows to find pairs of mutually inverse elements together with the calculation of $W$-orbits in the same runtime cycle. This simplifies the calculation of conjugacy classes in the Weyl group. As an example, the complete list of elements of the Weyl group $W(D_4)$ obtained using the extended Snow's algorithm is given. The elements of $W(D_4)$ are specified in two ways: as reduced expressions and as matrices of the faithful representation. We present a partition of this group into conjugacy classes with elements specified as reduced expressions. Various forms are given for representatives of the conjugacy classes of $W(D_4)$: using Carter diagrams, using reduced expressions and using signed cycle-types. In the appendix, we provide an implementation of the algorithm in Python.

math.RT↗

arXiv2022

Some approaches used to overcome overestimation in Deep Reinforcement Learning algorithms

Some phenomena related to statistical noise which have been investigated by various authors under the framework of deep reinforcement learning (RL) algorithms are discussed. The following algorithms are examined: the deep Q-network (DQN), double DQN, deep deterministic policy gradient (DDPG), twin-delayed DDPG (TD3), and hill climbing algorithm. First, we consider overestimation, which is a harmful property resulting from noise. Then we deal with noise used for exploration, this is the useful noise. We discuss setting the noise parameter in the TD3 for typical PyBullet environments associated with articulate bodies such as HopperBulletEnv and Walker2DBulletEnv. In the appendix, in relation to the hill climbing algorithm, another example related to noise is considered - an example of adaptive noise.

cs.LG↗

arXiv2014

Classification of linkage systems

A linkage diagram is obtained from the Carter diagram $Γ$ by adding an extra root $γ$, so that the resulting subset of roots is linearly independent. With every linkage diagram we associate the linkage label vector $γ^{\nabla}$, similar to Dynkin labels. The linkage diagrams connected under the action of the group $W^{\vee}_{S}$ constitute the the linkage system $\mathscr{L}(Γ)$. For any simply-laced Carter diagram, the system $\mathscr{L}(Γ)$ is constructed. To obtain linkage diagrams $θ^{\nabla}$, we use an easily verifiable criterion: $\mathscr{B}^{\vee}_Γ(θ^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_Γ$ is the inverse quadratic form associated with $Γ$. A Dynkin diagram $Γ'$ such that rank($Γ'$) = rank($Γ$) + 1 and any $Γ$-associated root subset $S$ lies in $\varPhi(Γ')$, is said to be the Dynkin extension. The linkage system $\mathscr{L}(Γ)$ is the union of $Γ_i$-components $\mathscr{L}_{Γ_i}(Γ)$ taken for all Dynkin extensions of $Γ<_D Γ_i$. The subset $\varPhi(S)$ of roots of $\varPhi(Γ')$, linearly dependent on roots of $S$ is said to be a partial root system. The size of $\mathscr{L}_{Γ'}(Γ)$ is estimated as follows: $|\mathscr{L}_{Γ'}(Γ)| \leq |\varPhi(Γ')| - |\varPhi(S)|$. Carter diagrams $E_l$ and $E_l(a_i)$ (resp. $D_l$ and $D_l(a_k)$) are said to be covalent. For any pair {$Γ, \widetildeΓ$} of covalent Carter diagrams, where $Γ$ is the Dynkin diagram, we explicitly construct the invertible linear map $M : \mathcal{P} \longrightarrow \mathcal{R}$, where $\mathcal{R}$ (resp. $\mathcal{P}$) is the root system (resp. partial root system) corresponding to $Γ$ (resp. $\widetildeΓ$). In particular, we have $|\mathscr{L}(\widetildeΓ)| = |\mathscr{L}(Γ)|$.

math.RT↗

arXiv2013

Coxeter Transformations, the McKay correspondence, and the Slodowy correspondence

This talk was presented at Workshop "Spectral Methods in Representation Theory of Algebras and Applications to the Study of Rings of Singularities", 2008 (Banff, Canada). W. Ebeling established a connection between certain Poincare series, the Coxeter transformation C, and the corresponding affine Coxeter transformation C_a (in the context of the McKay correspondence). We consider the generalized Poincare series [\tilde{P}_G(t)]_0 for the case of multiply-laced diagrams(in the context of the McKay-Slodowy correspondence) and extend the Ebeling theorem for this case: [\tilde{P}_G(t)]_0 = X(t^2)/\tilde{X}(t^2), where X is the characteristic polynomial of the Coxeter transformation and \tilde{X} is the characteristic polynomial of the corresponding affine Coxeter transformation. We obtain that Poincare series coincide for pairs of diagrams obtained by folding: X (Γ) / X (\tilde{Γ}) = X (Γ^f) / X (\tilde{Γ}^f), where Γ is any (A, D, E type) Dynkin diagram, Γ is the extended Dynkin diagram, and the diagrams Γ^f and \tilde{Γ}^f are obtained by folding from Γ and \tilde{Γ}, respectively.

math.RT↗

arXiv2012

Root systems and diagram calculus. I. Regular extensions of Carter diagrams and the uniqueness of conjugacy classes

In 1972, R. Carter introduced admissible diagrams to classify conjugacy classes in a finite Weyl group W. We say that an admissible diagram Γis a Carter diagram if any edge {α, β} with inner product (α, β) > 0 (resp. (α, β) < 0) is drawn as dotted (resp. solid) edge. We construct an explicit transformation of any Carter diagram containing long cycles (with the number of vertices l > 4) into another Carter diagram containing only 4-cycles. Thus, all Carter diagrams containing long cycles can be eliminated from the classification list. There exist diagrams determining two conjugacy classes in W.It is shown that any connected Carter diagram Γcontaining a 4-vertex pattern D_4 or D_4(a_1) determines a single conjugacy class. The main approach is studying different extensions of Carter diagrams. Let \tildeΓ be the Carter diagram obtained from a certain Carter diagram Γby adding a single vertex αconnected to Γat n points, n \leq 3. Let a socket be the set of vertices of Γconnected to α. If the number of sockets available for extensions is equal to 2, then there is a pair of extensions Γ< \tildeΓ_L and Γ< \tildeΓ_R, called mirror extensions and the pair elements w_L and w_R associated with \tildeΓ_L and \tildeΓ_R. We show that w_R = T^{-1}w_L{T} for some T \in W, where the map T is explicitly constructed for all mirror extensions. In Carter's description of the conjugacy classes in a Weyl group a key result (Carter's theorem) states that every element in a Weyl group is a product of two involutions. One of the goals of this paper and its sequels is to prepare the notions and framework in which we give the proof of this fact without appealing to the classification of conjugacy classes.

math.RT↗

arXiv2011

Root systems and diagram calculus. III. Semi-Coxeter orbits of linkage diagrams and the Carter theorem

A diagram obtained from the Carter diagram $Γ$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent is said to be the {\it linkage diagram}. Given a linkage diagram, we associate the linkage labels vector, which is introduced like the vector of Dynkin labels. Similarly to the dual Weyl group, we introduce the group $W^{\vee}_L$ associated with $Γ$, and we call it the dual partial Weyl group. The linkage labels vectors connected under the action of $W^{\vee}_L$ constitute the linkage system $\mathscr{L}(Γ)$, which is similar to the weight system arising in the representation theory of the semisimple Lie algebras. The Carter theorem states that every element of a Weyl group $W$ is expressible as the product of two involutions. We give the proof of this theorem based on the description of the linkage system $\mathscr{L}(Γ)$ and semi-Coxeter orbits of linkage labels vectors for any Carter diagram $Γ$. The main idea of the proof is based on the fact that, with a few exceptions, in each semi-Coxeter orbit there is a special linkage diagram -- called {\it unicolored}, for which the decomposition into the product of two involutions is trivial.

math.RT↗

arXiv2011

Root systems and diagram calculus. II. Quadratic forms for the Carter diagrams

For any Carter diagram $Γ$ containing 4-cycle, we introduce the partial Cartan matrix $B_L$, which is similar to the Cartan matrix associated with a Dynkin diagram. A linkage diagram is obtained from $Γ$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent. The linkage diagrams connected under the action of dual partial Weyl group (associated with $B_L$) constitute the linkage system, which is similar to the weight system arising in the representation theory of the semisimple Lie algebras. For Carter diagrams $E_6(a_i)$ and $E_6$ (resp. $E_7(a_i)$ and $E_7$; resp. $D_n(a_i)$ and $D_n$), the linkage system has, respectively, 2, 1, 1 components, each of which contains, respectively, 27, 56, $2n$ elements. Numbers 27, 56 and $2n$ are well-known dimensions of the smallest fundamental representations of semisimple Lie algebras, respectively, for $E_6$, $E_7$ and $D_n$. The 8-cell "spindle-like" linkage subsystems called loctets play the essential role in describing the linkage systems. It turns that weight systems also can be described by means of loctets.

math.RT↗

arXiv2009

The Poincare series of the hyperbolic Coxeter groups with finite volume of fundamental domains

The discrete group generated by reflections of the sphere, or Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series (a.k.a. growth function) is the generating function of the cardinalities of sets of elements of equal length. Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave a recurrence formula. The explicit expression of the Poincaré series was known for the spherical and Euclidean Coxeter groups, and 3-generated Coxeter groups, and (with mistakes) Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, and denominators for each (quasi-)Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the Poincaré series. The non-real poles behave as in Eneström's theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem's requirements.

math.RT↗

arXiv2006

Kostant's generating functions, Ebeling's theorem and McKay's observation relating the Poincare series

We generalize B. Kostant's construction of generating functions to the case of multiply-laced diagrams and we prove for this case W. Ebeling's theorem which connects the Poincare series [P_G(t)]_0 and the Coxeter transformations. According to W. Ebeling's theorem [P_G(t)]_0 = \frac{X(t^2)}{\tilde{X}(t^2)}, where X is the characteristic polynomial of the Coxeter transformation and \tilde{X} is the characteristic polynomial of the corresponding affine Coxeter transformation. We prove McKay's observation relating the Poincare series [P_G(t)]_i: (t+t^{-1})[P_G(t)]_i = \sum\limits_{i \leftarrow j}[P_G(t)]_j, where j runs over all vertices adjacent to i.

math.RT↗

arXiv2005

Notes on Coxeter Transformations and the McKay correspondence

We study in detail the Jordan forms of the Coxeter transformations and prove shearing formulas due to Subbotin and Sumin for the characteristic polynomials of the Coxeter transformations. Using shearing formulas we calculate characteristic polynomials of the Coxeter transformation for the diagrams T_{2,3,r}, T_{3,3,r}, T_{2,4,r}, prove J. S. Frame's formulas, and generalize R. Steinberg's theorem on the spectrum of the affine Coxeter transformation for the multiply-laced diagrams. This theorem is the key statement in R. Steinberg's proof of the McKay correspondence. B. Kostant's construction appears in the context of the McKay correspondence and gives a way to obtain multiplicities of indecomposable representations ρ_i of the binary polyhedral group G in the decomposition of π_n|G. In the case of multiply-laced graphs, instead of indecomposable representations ρ_i we use restricted representations and induced representations of G introduced by P. Slodowy. Using B. Kostant's construction we generalize to the case of multiply-laced graphs W. Ebeling's theorem which connects the Poincare series and the Coxeter transformations. Using the Jordan form of the Coxeter transformation we prove a criterion of V. Dlab and C. M. Ringel of regularity of quiver representations.

math.RT↗

arXiv2005

On Admissible and Perfect Elements in the Modular Lattice

For the modular lattice D^4 = {1+1+1+1} associated with the extended Dynkin diagram \tilde{D}_4 (and also for D^r, where r > 4), Gelfand and Ponomarev introduced the notion of admissible and perfect lattice elements and classified them. In this work, we classify the admissible and perfect elements in the modular lattice D^{2,2,2} = {2+2+2} associated with the extended Dynkin diagram \tilde{E}_6. Gelfand and Ponomarev constructed admissible elements for D^r recurrently in the length of multi-indices, which they called admissible sequences. Here we suggest a direct method for creating admissible elements. Admissible sequences and admissible elements for D^{2,2,2} (resp. D^4) form 14 classes (resp. 11 classes) and possess some periodicity. If under all indecomposable representations of a modular lattice the image of an element is either zero or the whole representation space, the element is said to be perfect. Our classification of perfect elements for D^{2,2,2} is based on the description of admissible elements. The constructed set H^+ of perfect elements is the union of 64-element distributive lattices H^+(n), and H^+ is the distributive lattice itself. The lattice of perfect elements B^+ obtained by Gelfand and Ponomarev for D^4 can be imbedded into the lattice of perfect elements H^+, associated with D^{2,2,2}.

math.RT↗