Jordan Type Research Articles

On $3$-generated axial algebras of Jordan type $\frac{1}{2}$

Axial algebras of Jordan type $\eta$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peirce decomposition for idempotents in Jordan algebras. A universal $3$-generated algebra of Jordan type $\frac{1}{2}$ as an algebra with $4$ parameters was constructed by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In this paper, we describe all semisimple $3$-generated algebras of Jordan type $\frac{1}{2}$ over a quadratically closed field.

Kirillov Polynomials for the Exceptional Lie Algebra g2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak g_{2}$$\\end{document}

As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra g2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak g_2$$\\end{document}. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.

On Jordan algebras that are factors of Matsuo algebras

We describe all finite connected 3-transposition groups whose Matsuo algebras have nontrivial factors that are Jordan algebras. As a corollary, we show that if $\mathbb{F}$ is a field of characteristic 0, then there exist infinitely many primitive axial algebras of Jordan type $\frac{1}{2}$ over $\mathbb{F}$ that are not factors of Matsuo algebras. As an example, we prove this for an exceptional Jordan algebra over~$\mathbb{F}$.

Modular representations in type A with a two-row nilpotent central character

We study the category of representations of slm+2n over a field of characteristic p with p≫0, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Hölder multiplicities of the simples inside the baby Vermas. We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the BMR equivalence. Our results generalize Jantzen's formulae in the subregular nilpotent case (i.e. when n=1), and may be viewed as a positive characteristic analogue of the combinatorial description for Kazhdan-Lusztig polynomials of Grassmannian permutations due to Lascoux and Schutzenberger.

Primitive 4-generated axial algebras of Jordan type

We show that primitive 4-generated axial algebras of Jordan type are at most 81-dimensional.

Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four

In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1,13,12,13,1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r=4, Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H-vector.The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H-vector fails to have WLP. In codimension c=3 it is conjectured that all AG rings have WLP. For c=4, Gondim shows in (Gondim, 2017) that WLP always holds for r≤4 and gives a family where WLP fails for any r≥7, building on Ikeda's example (Ikeda, 1996) of failure for r=5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c=4 and r≤6.

A novel adaptive approach for improvement in the estimation of hourly diffuse solar radiation: A case study of China

Liu & Jordan–type models are still widely used to estimate diffuse radiation in various regions because of their simplicity. This paper proposes an adaptive approach of combining the grouping of kt–k data points with Liu & Jordan–type models to estimate the hourly diffuse radiation more accurately. This study was conducted using hourly radiation data measured in three regions in China (Shanghai, Xi'an, and Lhasa). The main novelties reported herein are the introduction of α as a constraint indicator and the proposal of a criterion for grouping kt–k data points. Grouping models, which adopted the functional form of cubic polynomials, were established in three regions based on the grouped data. Compared with the all-data model results, the percentage of mean absolute bias error (MABE%) and relative root mean square error (RMSE%) for the grouping model decreased by approximately 4% in Shanghai and Xi'an and 6% in Lhasa. Furthermore, weather condition analysis showed that the grouping models had the best adaptability on overcast days, followed by cloudy days and sunny days. A similar analysis was performed for solar time, with the best adaptability occurring at sunrise and sunset and the worst occurring at 9 am. The results also suggest that similar working procedures could contribute to hourly diffuse radiation estimation in other fields.

Structure of primitive axial algebras

A “fusion law” is a collection of multiplication rules among eigenspaces of an idempotent. This terminology is relatively new and is closely related to primitive axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras are closely related to 3-transposition groups and vertex operator algebras.In earlier work we studied primitive axial algebras, not necessarily commutative, and showed that they all have Jordan type. In this paper, we show that all finitely generated primitive axial algebras are direct sums of specifically described flexible finite-dimensional noncommutative algebras, and commutative axial algebras generated by primitive axes of the same type. In particular, all primitive axial algebras are flexible. They also have Frobenius forms. We give a precise description of all the primitive axes of axial algebras generated by two primitive axes.

Number of generators of ideals in Jordan cells of the family of graded Artinian algebras of height two

We let A=R/I be a standard graded Artinian algebra quotient of R=k[x,y], the polynomial ring in two variables over a field k by an ideal I, and let n be its vector space dimension. The Jordan type Pℓ of a linear form ℓ∈A1 is the partition of n determining the Jordan block decomposition of the multiplication on A by ℓ – which is nilpotent. The first three authors previously determined which partitions of n=dimk⁡A may occur as the Jordan type for some linear form ℓ on a graded complete intersection Artinian quotient A=R/(f,g) of R, and they counted the number of such partitions for each complete intersection Hilbert function T[1].We here consider the family GT of graded Artinian quotients A=R/I of R=k[x,y], having arbitrary Hilbert function H(A)=T. The Jordan cell V(EP) corresponding to a partition P having diagonal lengths T is comprised of all ideals I in R whose initial ideal is the monomial ideal EP determined by P. These cells give a decomposition of the variety GT into affine spaces. We determine the generic number κ(P) of generators for the ideals in each cell V(EP), generalizing a result of [1]. In particular, we determine those partitions for which κ(P)=κ(T), the generic number of generators for an ideal defining an algebra A in GT. We also count the number of partitions P of diagonal lengths T having a given κ(P). A main tool is a combinatorial and geometric result allowing us to split T and any partition P of diagonal lengths T into simpler Ti and partitions Pi, such that V(EP) is the product of the cells V(EPi), and Ti is single-block: GTi is a Grassmannian.

Varieties of subalgebras and endotrivial modules

Let (g,[p]) be a finite dimensional restricted Lie algebra over a perfect field k of characteristic p≥3. By combining methods from recent work of Benson-Carlson [4] with those of [10,16] we obtain a description of the endotrivial (g,[p])-modules in case the underlying Lie algebra g is supersolvable. For such g and algebraically closed k, this yields a classification of the indecomposable (g,[p])-modules of constant Jordan type with one non-projective block.

Pseudo-quotients of algebraic actions and their application to character varieties

In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good Geometric Invariant Theory (GIT) quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of [Formula: see text]-character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type.

Primitive axial algebras are of Jordan type

The notion of axial algebra is closely related to 3-transposition groups, the Monster group and vertex operator algebras. In this work we continue our previous works and compete the proof that all algebras generated by a set of primitive axes not necessarily of the same type (see the definition in the body of the paper), are primitive axial algebras of Jordan type.

Intersection cohomology of character varieties for punctured Riemann surfaces

We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on a previous result from Mellit [Mel20a] for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier [Let15].

Sharped Jordan's type inequalities with exponential approximations

Sharped Jordan's type inequalities with exponential approximations

Classification of the pairs of matrices of fixed Jordan types and representations of bundles of semichains

We study the problem of classifying the pairs of matrices over an algebraically closed field with some restrictions on their Jordan canonical forms using earlier results of the first author. All tame and wild, polynomial and non-polynomial growth cases are described.

Multiplicative Jordan type higher derivations of unital rings with non trivial idempotents

Multiplicative Jordan type higher derivations of unital rings with non trivial idempotents

Exotic t-structures for two-block Springer fibres

We study the category of representations of s l m + 2 n \mathfrak {sl}_{m+2n} in positive characteristic, with p p -character given by a nilpotent with Jordan type ( m + n , n ) (m+n,n) . Recent work of Bezrukavnikov-Mirkovic [Ann. of Math. (2) 178 (2013), pp. 835–919] implies that this representation category is equivalent to D m , n 0 \mathcal {D}_{m,n}^0 , the heart of the exotic t-structure on the derived category of coherent sheaves on a Springer fibre for that nilpotent. Using work of Cautis and Kamnitzer [Duke Math. J. 142 (2008), pp. 511–588], we construct functors indexed by affine tangles, between these categories D m , n \mathcal {D}_{m,n} (i.e. for different values of n n ). This allows us to describe the irreducible objects in D m , n 0 \mathcal {D}_{m,n}^0 and enumerate them by crossingless ( m , m + 2 n ) (m,m+2n) matchings. We compute the E x t \mathrm {Ext} spaces between the irreducible objects, and conjecture that the resulting Ext algebra is an annular variant of Khovanov’s arc algebra. In subsequent work, we use these results to give combinatorial dimension formulae for the irreducible representations. These results may be viewed as a positive characteristic analogue of results about two-block parabolic category O \mathcal {O} due to Lascoux-Schutzenberger [Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266], Bernstein-Frenkel-Khovanov [Selecta Math. (N.S.) 5 (1999), pp. 199–241], Brundan-Stroppel [Represent. Theory 15 (2011), pp. 170–243], et al.

The relations between the von Neumann–Jordan type constant and some geometric properties of Banach spaces

The relations between the von Neumann–Jordan type constant and some geometric properties of Banach spaces

Artinian algebras and Jordan type

There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Artinian Gorenstein. A more general invariant of an Artinian algebra A or finite A-module M that we consider here is the set of Jordan types of elements of the maximal ideal 𝔪 of A, acting on M. Here, the Jordan type of ℓ∈𝔪A is the partition giving the Jordan blocks of the multiplication map mℓ:M→M. In particular, we consider the Jordan type of a generic linear element ℓ in A1, or in the case of a local ring 𝒜, that of a generic element ℓ∈𝔪𝒜, the maximum ideal. We often take M=A, the graded algebra, or M=𝒜 a local algebra. The strong Lefschetz property of an element, as well as the weak Lefschetz property can be expressed simply in terms of its Jordan type and the Hilbert function of M. However, there has not been until recently a systematic study of the set of possible Jordan types for a given Artinian algebra A or A-module M, except, importantly, in modular invariant theory, or in the study of commuting Jordan types. We first show some basic properties of the Jordan type. In a main result we show an inequality between the Jordan type of ℓ∈𝔪𝒜 and a certain local Hilbert function. In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We also propose open problems.

Jordan Types of Triangular Matrices over a Finite Field

Let \(\lambda \) be a partition of an integer n and \({\mathbb F}_q\) be a finite field of order q. Let \(P_\lambda (q)\) be the number of strictly upper triangular \(n\times n\) matrices of the Jordan type \(\lambda \). It is known that the polynomial \(P_\lambda \) has a tendency to be divisible by high powers of q and \(Q=q-1\), and we put \(P_\lambda (q)=q^{d(\lambda )}Q^{e(\lambda )}R_\lambda (q)\), where \(R_\lambda (0)\ne 0\) and \(R_\lambda (1)\ne 0\). In this article, we study the polynomials \(P_\lambda (q)\) and \(R_\lambda (q)\). Our main results: an explicit formula for \(d(\lambda )\) (an explicit formula for \(e(\lambda )\) is known, see Proposition 3.3 below), a recursive formula for \(R_\lambda (q)\) (a similar formula for \(P_\lambda (q)\) is known, see Proposition 3.1 below), the stabilization of \(R_\lambda \) with respect to extending \(\lambda \) by adding strings of 1’s, and an explicit formula for the limit series \(R_{\lambda 1^\infty }\). Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.