Canonical Basis Research Articles

Pattern groups and a poset based Hopf monoid

The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

A Fock space model for decomposition numbers for quantum groups at roots of unity

In this paper we construct an "abstract Fock space" for general Lie types that serves as a generalisation of the infinite wedge $q$-Fock space familiar in type $A$. Specifically, for each positive integer $\ell$, we define a $\mathbb{Z}[q,q^{-1}]$-module $\mathcal{F}_{\ell}$ with bar involution by specifying generators and "straightening relations" adapted from those appearing in the Kashiwara-Miwa-Stern formulation of the $q$-Fock space. By relating $\mathcal{F}_{\ell}$ to the corresponding affine Hecke algebra we show that the abstract Fock space has standard and canonical bases for which the transition matrix produces parabolic affine Kazhdan-Lusztig polynomials. This property and the convenient combinatorial labeling of bases of $\mathcal{F}_{\ell}$ by dominant integral weights makes $\mathcal{F}_{\ell}$ a useful combinatorial tool for determining decomposition numbers of Weyl modules for quantum groups at roots of unity.

The Council of Metropolitan See and Its Canonical Basis. An Orthodox Approach

In this canonical study, the reader of the article will have an opportunity to become acquainted with an old European canonical-juridical institution, that is, the synodality, and, ipso facto, about its juridical regime. And, naturally, the reader can get acquainted more closely with the provisions of the canonical norms of the Eastern Church regarding the eparchial (metropolitan) synodality institution, and, ipso facto, with the issue of the regime of the synodality. Since the canonical bases of the synodality regime are foreseen in the canonical legislation of the first millennium, we had to make an hermeneutical analysis of its text, which showed us that, by resorting to ad fontes, we can also pave the way that would go towards the restoration of the unity of the two Christian worlds, namely, Pars Orientis and Pars Occidentis.

Diagram automorphisms and quantum groups

Let $\mathbf{U}^-_q = \mathbf{U}^-_q(\mathfrak{g})$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, and $\sigma : \mathfrak{g} \to \mathfrak{g}$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak{g}^{\sigma}$ be the fixed point subalgebra of $\mathfrak{g}$, and put $\underline{\mathbf{U}}^-_q = \mathbf{U}^-_q(\mathfrak{g}^{\sigma})$. Let $\mathbf{B}$ be the canonical basis of $\mathbf{U}_q^-$ and $\underline{\mathbf{B}}$ the canonical basis of $\underline{\mathbf{U}}_q^-$. $\sigma$ induces a natural action on $\mathbf{B}$, and we denote by $\mathbf{B}^{\sigma}$ the set of $\sigma$-fixed elements in $\mathbf{B}$. Lusztig proved that there exists a canonical bijection $\mathbf{B}^{\sigma} \simeq \underline{\mathbf{B}}$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBW-bases constructed by Beck and Nakajima.

Discrete antiderivatives for functions over mathop {{mathbb {F}}}_p^n

In the design of cryptographic functions, the properties of their discrete derivatives have to be carefully considered, as many cryptographic attacks exploit these properties. One can therefore attempt to first construct derivatives with the desired properties and then recover the function itself. Recently Suder developed an algorithm for reconstructing a function (also called antiderivative) over the finite field mathop {{mathbb {F}}}_{2^n} given its discrete derivatives in up to n linearly independent directions. Pasalic et al. also presented an algorithm for determining a function over mathop {{mathbb {F}}}_{p^n} given one of its derivatives. Both algorithms involve solving a p^n times p^n system of linear equations; the functions are represented as univariate polynomials over mathop {{mathbb {F}}}_{p^n}. We show that this apparently high computational complexity is not intrinsic to the problem, but rather a consequence of the representation used. We describe a simpler algorithm, with quasilinear complexity, provided we work with a different representation of the functions. Namely they are polynomials in n variables over mathop {{mathbb {F}}}_{p} in algebraic normal form (for p>2, additionally, we need to use the falling factorial polynomial basis) and the directions of the derivatives are the canonical basis of mathop {{mathbb {F}}}_{p}^n. Algorithms for other representations (the directions of the derivatives not being the canonical basis vectors or the univariate polynomials over mathop {{mathbb {F}}}_{p^n} mentioned above) can be obtained by combining our algorithm with converting between representations. However, the complexity of these conversions is, in the worst case, exponential. As an application, we develop a method for constructing new quadratic PN (Perfect Nonlinear) functions. We use an approach similar to the one of Suder, who used antiderivatives to give an alternative formulation of the methods of Weng et al. and Yu et al. for searching for new quadratic APN (Almost Perfect Nonlinear) functions.

Categorified canonical bases and framed BPS states

We consider a cluster variety associated to a triangulated surface without punctures. The algebra of regular functions on this cluster variety possesses a canonical vector space basis parametrized by certain measured laminations on the surface. To each lamination, we associate a graded vector space, and we prove that the graded dimension of this vector space gives the expansion in cluster coordinates of the corresponding basis element. We discuss the relation to framed BPS states in $\mathcal{N}=2$ field theories of class $\mathcal{S}$.

Generating formulas for finite reflection groups of the infinite series $$S_n$$, $$A_n$$, $$B_n$$ and $$D_n$$

Let $W$ be a rank $n$ irreducible finite reflection group and let $p_1(x),\ldots,p_n(x)$, $x\in\mathbb{R}^n$, be a basis of algebraically independent $W$-invariant real homogeneous polynomials. The orbit map $\overline p:\mathbb{R}^n\to\mathbb{R}^n:x\to (p_1(x),\ldots,p_n(x))$ induces a diffeomorphism between the orbit space $\mathbb{R}^n/W$ and the set ${\cal S}=\overline p(\mathbb{R}^n)\subset\mathbb{R}^n$. The border of ${\cal S}$ is the $\overline p$ image of the set of reflecting hyperplanes of $W$. With a given basic set of invariant polynomials it is possible to build an $n\times n$ polynomial matrix, $\widehat P(p)$, $p\in\mathbb{R}^n$, sometimes called $\widehat P$-matrix, such that $\widehat P_{ab}(p(x))=\nabla p_a(x)\cdot \nabla p_b(x)$, $\forall\,a,b=1,\ldots,n$. The border of ${\cal S}$ is contained in the algebraic surface $\det(\widehat P(p))=0$, sometimes called discriminant, and the polynomial $\det(\widehat P(p))$ satisfies a system of differential equations that depends on an $n$-dimensional polynomial vector $\lambda(p)$. Possible applications concern phase transitions and singularities. If the rank $n$ is large, the matrix $\widehat P(p)$ is in general difficult to calculate. In this article I suggest a choice of the basic invariant polynomials for all the reflection groups of type $S_n$, $A_n$, $B_n$, $D_n$, $\forall\,n\in \mathbb{N}$, for which I give generating formulas for the corresponding $\widehat P$-matrices and $\lambda$-vectors. These $\widehat P$-matrices can be written, almost completely, as sums of block Hankel matrices. Transformation formulas allow to determine easily both the $\widehat P$-matrix and the $\lambda$-vector in any other basis of invariant polynomials. Examples of transformations into flat bases, $a$-bases, and canonical bases, are considered.

On canonical bases for the Letzter algebra [formula omitted

A canonical basis was constructed by Wang and the author in [6] inside Letzter's coideal subalgebra in quantum sl2. In this article, we provide an explicit description for the canonical bases and show that the bases coincide with the one defined algebraically by Bao-Wang in [1].

Uglov Bipartitions and Extended Young Diagrams

We study the class of Uglov bipartitions and prove a generalization of a conjecture by Dipper, James and Murphy. We give two consequences concerning the computation of canonical bases in affine type $A$ and the description of decomposition matrices for Hecke algebras of type $B_n$ in arbitrary characteristic.

Docking of Cisplatin on Fullerene Derivatives and Some Cube Rhombellane Functionalized Homeomorphs

Cisplatin (cisPt) is one of the strongest anticancer agents with proven clinical activity against a wide range of solid tumors. Its mode of action has been linked to its ability to crosslink with the canonical purine bases, primarily with guanine. Theoretical studies performed at the molecular level suggest that such nonspecific interactions can also take place with many competitive compounds, such as vitamins of the B group, containing aromatic rings with lone-pair orbitals. This might be an indicator of reduction of the anticancer therapeutic effects of the Cisplatin drug in the presence of vitamins of the B group inside the cell nucleus. That is why it seems to be important to connect CisPt with nanostructures and in this way prevent the drug from combining with the B vitamins. As a proposal for a new nanodrug, an attempt was made to implement Cispaltin (CisPt) ligand on functionalized C60 fullerenes and on a cube rhombellane homeomorphic surface. The symmetry of the analyzed nanostructures is an important factor determining the mutual affinity of the tested ligand and nanocarriers. The behavior of Cisplatin with respect to rhombellane homeomorphs and functionalized fullerenes C60, in terms of their (interacting) energy, geometry and topology was studied and a detailed analysis of structural properties after docking showed many interesting features.

Schur Algebras and Quantum Symmetric Pairs With Unequal Parameters

Abstract We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson–Lusztig–MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type AIII/AIV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated with any weight function. These bases are the counterparts of Lusztig’s bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson–Lusztig–MacPherson construction, which is first introduced by Fan–Li from a geometric viewpoint.

Derivations and dimensionally nilpotent derivations in Lie triple algebras

In this paper, we first study derivations in non nilpotent Lie triple algebras. We determine the structure of derivation algebra according to whether it admits an idempotent or a pseudo-idempotent. We study the multiplicative structure of non nil dimensionally nilpotent Lie triple algebras. We show that when n=2p+1 the adapted basis coincides with the canonical basis of the gametic algebra G(2p+2,2) or this one obviously associated to a pseudo-idempotent and if n=2p then the algebra is either one of the precedent case or a conservative Bernstein algebra.

Holes and dependences in an ordered complex

We use the canonical bases produced by the tri-partition algorithm in (Edelsbrunner and Ölsböck, 2018) to open and close holes in a polyhedral complex, K. In a concrete application, we consider the Delaunay mosaic of a finite set, we let K be an Alpha complex, and we use the persistence diagram of the distance function to guide the hole opening and closing operations. The dependences between the holes define a partial order on the cells in K that characterizes what can and what cannot be constructed using the operations. The relations in this partial order reveal structural information about the underlying filtration of complexes beyond what is expressed by the persistence diagram.

Epigenetic Methylations on N6-Adenine and N6-Adenosine with the same Input but Different Output.

Epigenetic modifications on individual bases in DNA and RNA can encode inheritable genetic information beyond the canonical bases. Among the nucleic acid modifications, DNA N6-methadenine (6mA) and RNA N6-methyladenosine (m6A) have recently been well-studied due to the technological development of detection strategies and the functional identification of modification enzymes. The current findings demonstrate a wide spectrum of 6mA and m6A distributions from prokaryotes to eukaryotes and critical roles in multiple cellular processes. It is interesting that the processes of modification in which the methyl group is added to adenine and adenosine are the same, but the outcomes of these modifications in terms of their physiological impacts in organisms are quite different. In this review, we summarize the latest progress in the study of enzymes involved in the 6mA and m6A methylation machinery, including methyltransferases and demethylases, and their functions in various biological pathways. In particular, we focus on the mechanisms by which 6mA and m6A regulate the expression of target genes, and we highlight the future challenges in epigenetic regulation.

Affine cluster monomials are generalized minors

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with$\mathbf{g}$-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type$A_{1}^{(1)}$, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.

Resonant Mirković–Vilonen polytopes and formulas for highest-weight characters

Formulas for the product of an irreducible character $$\chi _\lambda $$ of a complex Lie group and a deformation of the Weyl denominator as a sum over the crystal $${\mathcal {B}}(\lambda +\rho )$$ go back to Tokuyama. We study the geometry underlying such formulas using the expansion of spherical Whittaker functions of p-adic groups as a sum over the canonical basis $${\mathcal {B}}(-\infty )$$ , which we show may be understood as arising from tropicalization of certain toric charts that appear in the theory of total positivity and cluster algebras. We use this to express the terms of the expansion in terms of the corresponding Mirkovic–Vilonen polytope. In this non-archimedean setting, we identify resonance as the appropriate analogue of total positivity, and introduce resonant Mirkovic–Vilonen polytopes as the corresponding geometric context. Focusing on the exceptional group $$G_2$$ , we show that these polytopes carry new crystal graph structures which we use to compute a new Tokuyama-type formula as a sum over $${\mathcal {B}}(\lambda +\rho )$$ plus a geometric error term coming from finitely many crystals of resonant polytopes.

The comultiplication of modified quantum affine 𝔰𝔩n

Let [Formula: see text] be the modified quantum affine [Formula: see text] and let [Formula: see text] be the positive part of quantum affine [Formula: see text]. Let [Formula: see text] be the canonical basis of [Formula: see text] and let [Formula: see text] be the canonical basis of [Formula: see text]. In this paper, we use the theory of affine quantum Schur algebras to prove that the structure constants for the comultiplication with respect to [Formula: see text] are determined by the structure constants for the comultiplication with respect to [Formula: see text] for [Formula: see text]. In particular, from the positivity property for the comultiplication of [Formula: see text], we obtain the positivity property for the comultiplication of [Formula: see text], which is conjectured by Lusztig [Introduction to Quantum Groups, Progress in Mathematics, Vol. 110 (Birkhäuser, Boston, 1993), 25.4.2].

Persuading Risk-Conscious Agents: A Geometric Approach

We consider a persuasion problem between a sender and a non-expected utility maximizing receiver whose utility may be nonlinear in her belief; we call such receivers risk-conscious. Such utility models arise, for example, when the receiver exhibits sensitivity to the variance of the payoff on choosing an action (e.g., uncertainty-aversion when waiting for a service). Due to this nonlinearity, the revelation principle fails and action recommendations no longer suffice for optimal persuasion. To overcome this challenge, we develop an optimization framework using the underlying geometry of the persuasion problem to pose it as a convex optimization program. We use this approach to analyze the setting of binary persuasion, where the receiver has two actions and the sender prefers one of them over the other in every state. Under a mild convexity assumption, we reduce the persuasion problem to a linear program, and establish a canonical basis for the set of signals in an optimal signaling scheme. The signals in this canonical set either reveal the state, or induce in the receiver uncertainty between two states. Finally, we apply our methods to optimally share waiting time information in a queueing system with uncertainty-averse customers.

Mathematical and visual models of asynchronous electric machines energy fields

The research considers the development of topological theory of asynchronous electric machines (AEM). The results of the development of energy aspects of the topological theory are presented in the article. The facts proving the actuality of the research are also given. The central notion giving the topological theory the characteristic of being systematic is the vector space notion. The definition of AEM vector space, its dimension, canonical basis, power and energy field of AEM are presented. Mathematical and visual models of AEM energy fields are analyzed. Energy fields of electric losses and magnetic field dissipation having the spherical symmetry are analyzed in particular. Energy and visual models of the main magnetic field are analyzed using AEM with a short circuited rotor and linear AEM as examples. It is established that this field obtains cylindrical or elliptical symmetry. In general the main magnetic field and the main matrix of winding make AEM vector space fully or partially heterogeneous. Energy efficiency is analyzed and its differences for vector space currents are revealed. The areas of vector spaces with high and low energy currents efficiency are defined. According to this indicator longitudinal and transvers subsets of current having, correspondingly, high and zero energy efficiency are distinguished in the vector space. The description of practical application spheres of AEM topological theory is given.

Cyclic Demazure Modules and Positroid Varieties

A positroid variety is an intersection of cyclically rotated Grassmannian Schubert varieties. Each graded piece of the homogeneous coordinate ring of a positroid variety is the intersection of cyclically rotated (rectangular) Demazure modules, which we call the cyclic Demazure module. In this note, we show that the cyclic Demazure module has a canonical basis, and define the cyclic Demazure crystal.