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The p-standard table algebras of order p 3 for any rational prime p, which include the adjacency algebras of p-schemes of order p 3 , are classified, provided that the thin radical of the distinguished basis of the table algebra has order p. The classification is in terms of wreath products, and of certain vectors (that we call Legendre elements) in the group algebra of a group of order p. Their coefficients yield real solutions to the classical equations that hold for the Legendre symbols. A fully explicit classification of the relevant Legendre elements, and hence of the table algebras at issue, is attained in the case of integer structure constants, as we characterize certain real solutions to these equations sufficiently enough to show that the only integer solutions are in fact ± the Legendre symbols themselves and appropriate translates thereof. Criteria for two such table algebras being isomorphic are also established in this case, and the automorphism group of any one such table algebra is determined.

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of X are uniquely determined by Witten's presentation of QH(X) and the fact that they are non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, this conjecture is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton., 16 pages

In this paper, we study nilpotent Lie algebras that admit nilsoliton metric with simple pre-Einstein derivation. Given a Lie algebra η , we would like to compute as much of its structure as possible. The structural components we consider in this study are the structure constants, the index, and the rank of the nilsoliton derivations. For this purpose, we prove necessary or sufficient conditions for an algebra to admit such metrics. Particularly, we prove theorems for the computation of the Jacobi identity for a given algebra so that we can solve the system of the equation(s) and find the structure constants of the nilsoliton.

We find the abnormal extremals on four-dimensional connected Lie groups with left-invariant sub-Finsler quasimetric defined by a seminorm on a two-dimensional subspace of the Lie algebra generating the algebra. In terms of the structure constants of a Lie algebra and the Minkowski support function of the unit ball of the seminorm on the two-dimensional subspace of a Lie algebra which defines a quasimetric, we establish a criterion for the strict abnormality of these extremals.

We investigate the structure constants of the center H n of the group algebra Z [ S p n ( q ) ] over a finite field. The reflection length on the group G L 2 n ( q ) induces a filtration on the algebras H n . We prove that the structure constants of the associated graded algebra S n are independent of n. As tool in the proof we consider the embedding S p m ↪ S p n ( q ) and determine the behavior of the centralizers and intersection of centralizers under the embedding S p m ↪ S p n ( q ) . We determine explicit formulas for rate of the growth of the centralizer of an element U ∈ S p m ( q ) in terms of dimension of the fixed space of U.

We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that the structure constants of the Kazhdan-Lusztig basis are unimodal. We explain why the relative hard Lefschetz theorem implies that the tensor category associated by Lusztig to any 2-sided cell in a Coxeter group is rigid and pivotal., 29 pages, v2: added a missing assumption in Remark 5.1

We consider an evolution algebra identifying the coefficients of SIS–SIR worm propagation models as the structure constants of the algebra. The basic properties of this algebra are studied. We prov...

When D is a linear partial differential operator of any order, a direct problem is to look for an operator D1 generating the compatibility conditions (CC) D1η = 0 of Dξ = η. Conversely, when D1 is given, an inverse problem is to look for an operator D such that its CC are generated by D1 and we shall say that D1 is parametrized by D = D0. We may thus construct a differential sequence with successive operators D, D1, D2, ..., each operator parametrizing the next one. Introducing the formal adjoint ad() of an operator, we have but ad (Di-1) may not generate all the CC of ad (Di). When D = K [d1, ..., dn] = K [d] is the (non-commutative) ring of differential operators with coefficients in a differential field K, then D gives rise by residue to a differential module M over D while ad (D) gives rise to a differential module N =ad (M) over D. The differential extension modules with ext0(M) = homD (M, D) only depend on M and are measuring the above gaps, independently of the previous differential sequence, in such a way that ext1 (N) = t (M) is the torsion submodule of M. The purpose of this paper is to compute them for certain Lie operators involved in the theory of Lie pseudogroups in arbitrary dimension n and to prove for the first time that the extension modules highly depend on the Vessiot structure constants c. Comparing the last invited lecture published in 1962 by Lanczos with a commutative diagram that we provided in a recent paper on gravitational waves, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. We shall prove that Lanczos was not trying to parametrize the Riemann operator but its formal adjoint Beltrami = ad (Riemann) which can indeed be parametrized by the operator Lanczos = ad (Bianchi) in arbitrary dimension, “one step further on to the right” in the Killing sequence. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the theory of Lie pseudogroups through double differential duality and the construction of finite length differential sequences for Lie operators. In particular, when one is dealing with a Lie group of transformations or, equivalently, when D is a Lie operator of finite type, we shall prove that . It will follow that the Riemann-Lanczos and Weyl-Lanczos problems just amount to prove such a result for i = 1,2 and arbitrary n when D is the classical or conformal Killing operator. We provide a description of the potentials allowing to parametrize the Riemann and the Weyl operators in arbitrary dimension, both with their adjoint operators. Most of these results are new and have been checked by means of computer algebra.

We completely determine the structure constants between real root vectors in a rank 2 Kac–Moody algebra g . Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the root lattice grow exponentially with height. Our approach is to extend Carter's method of finding structure constants from those on extraspecial pairs to the rank 2 Kac–Moody case. We also determine all commutator relations involving only real root vectors in all rank 2 Kac-Moody algebras. The generalized Cartan matrix of g is of the form H ( a , b ) = ( 2 − b − a 2 ) where a , b ∈ Z and a b ≥ 4 . If a b = 4 , then g is of affine type. If a b > 4 , then g is of hyperbolic type. Explicit knowledge of the root strings is needed, as well as a characterization of the pairs of real roots whose sums are real. We prove that if a and b are both greater than one, then no sum of real roots can be a real root. We determine the root strings between real roots β , γ in H ( a , 1 ) , a ≥ 5 and we determine the sets ( Z ≥ 0 α + Z ≥ 0 β ) ∩ Δ re ( H ( a , b ) ) . One of our tools is a characterization of the root subsystems generated by a subset of roots. We classify these subsystems in rank 2 Kac–Moody root systems. We prove that every rank two infinite root system contains an infinite family of non-isomorphic symmetric rank 2 hyperbolic root subsystems H ( k , k ) for certain k ≥ 3 , generated by either two short or two long simple roots. We also prove that a non-symmetric hyperbolic root systems H ( a , b ) with a ≠ b and a b > 5 also contains an infinite family of non-isomorphic non-symmetric rank 2 hyperbolic root subsystems H ( a l , b l ) , for certain positive integers l.

We first prove an identity involving symmetric polynomials. This identity leads us into exploring the geometry of Lagrangian Grassmannians. As an insight applications, we obtain a formula for the integral over the Lagrangian Grassmannian of a characteristic class of the tautological sub-bundle. Moreover, a relation to that over the ordinary Grassmannian and its application to the degree formula for the Lagrangian Grassmannian are given. Finally, we present further applications to the computation of Schubert structure constants and three-point, degree 1, genus 0 Gromov–Witten invariants of the Lagrangian Grassmannian. Some examples together with explicit computations are presented.

We derive a recursive formula for the structure constants for the universal enveloping algebra of 𝔰𝔩2(𝕂), where 𝕂 is an algebraically closed field of characteristic zero.

Two dimensional field theories invariant under the Bondi-Metzner-Sachs (BMS) group are conjectured to be dual to asymptotically flat spacetimes in three dimensions. In this paper, we continue our investigations of the modular properties of these field theories. In particular, we focus on the BMS torus one-point function. We use two different methods to arrive at expressions for asymptotic structure constants for general states in the theory utilising modular properties of the torus one-point function. We then concentrate on the BMS highest weight representation, and derive a host of new results, the most important of which is the BMS torus block. In a particular limit of large weights, we derive the leading and sub-leading pieces of the BMS torus block, which we then use to rederive an expression for the asymptotic structure constants for BMS primaries. Finally, we perform a bulk computation of a probe scalar in the background of a flatspace cosmological solution based on the geodesic approximation to reproduce our field theoretic results., Comment: 101 pages, 4 figures, 12 appendices; v2: typos and minor errors corrected, references added, acknowledgement updated, matches journal version

We explore geometries that give rise to a novel algebraic structure, the Exceptional Drinfeld Algebra, which has recently been proposed as an approach to study generalised U-dualities, similar to the non-Abelian and Poisson-Lie generalisations of T-duality. This algebra is generically not a Lie algebra but a Leibniz algebra, and can be realised in exceptional generalised geometry or exceptional field theory through a set of frame fields giving a generalised parallelisation. We provide examples including "three-algebra geometries", which encode the structure constants for three-algebras and in some cases give novel uplifts for $CSO(p,q,r)$ gaugings of seven-dimensional maximal supergravity. We also discuss the M-theoretic embedding of both non-Abelian and Poisson-Lie T-duality., 24 pages + appendices + refs. v2: published version

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we generalized to $\text{GL}_{n}$ in the paper "On graphs of Hecke operators". After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann and Fratila to develop an algorithm which explicitly calculate these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section., Comment: 38 pages, comments are welcome - minor typos were corrected in the second version

Let $\mathfrak{g}(A)$ be the Kac-Moody algebra with respect to a symmetrizable generalized Cartan matrix $A$. We give an explicit presentation of the fix-point Lie subalgebra $\mathfrak{k}(A)$ of $\mathfrak{g}(A)$ with respect to the Chevalley involution. It is a presentation of $\mathfrak{k}(A)$ involving inhomogeneous versions of the Serre relations, or, from a different perspective, a presentation generalizing the Dolan-Grady presentation of the Onsager algebra. In the finite and untwisted affine case we explicitly compute the structure constants of $\mathfrak{k}(A)$ in terms of a Chevalley type basis of $\mathfrak{k}(A)$. For the symplectic Lie algebra and its untwisted affine extension we explicitly describe the one-dimensional representations of $\mathfrak{k}(A)$., 20 pages. v2: typos corrected and references added

We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of the action of the alternating group $A_d$ on the $d$-fold tensor power of an $n$-dimensional $F$-vector space. When $F$ has characteristic different from $2$, we give a basis of $AS_F(n,d)$ in terms of bipartite graphs, and a graphical interpretation of the structure constants. We introduce the abstract Koszul duality functor on modules for the even part of any $\mathbf Z/2\mathbf Z$-graded algebra. The algebra $AS_F(n,d)$ is $\mathbf Z/2\mathbf Z$-graded, having the classical Schur algebra $S_F(n,d)$ as its even part. This leads to an approach to Koszul duality for $S_F(n,d)$-modules that is amenable to combinatorial methods. We characterize the category of $AS_F(n,d)$-modules in terms of $S_F(n,d)$-modules and their Koszul duals. We use the graphical basis of $AS_F(n,d)$ to study the dependence of the behavior of derived Koszul duality on $n$ and $d$., 27 pages, 1 figure, to appear in Pacific Journal of Mathematics

We study fermionic conformal field theories on surfaces with spin structure in the presence of boundaries, defects, and interfaces. We obtain the relevant crossing relations, taking particular care with parity signs and signs arising from the change of spin structure in different limits. We define fermionic classifying algebras for boundaries, defects, and interfaces, which allow one to read off the elementary boundary conditions, etc. As examples, we define fermionic extensions of Virasoro minimal models and give explicit solutions for the spectrum and bulk structure constants. We show how the $A$- and $D$-type fermionic Virasoro minimal models are related by a parity-shift operation which we define in general. We study the boundaries, defects, and interfaces in several examples, in particular in the fermionic Ising model, i.e. the free fermion, in the fermionic tri-critical Ising model, i.e. the first unitary $N=1$ superconformal minimal model, and in the supersymmetric Lee-Yang model, of which there are two distinct versions that are related by parity-shift., Comment: 40 pages, 3 figures

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions $\Lambda$ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric functions are defined as weighted sums over cylindric reverse plane partitions or - alternatively - in terms of sets of affine permutations. We relate their combinatorial definition to an algebraic construction in terms of the principal Heisenberg subalgebra of the affine Lie algebra $\mathfrak{\widehat{sl}}_n$ and a specialised cyclotomic Hecke algebra. Using Schur-Weyl duality we show that the new cylindric symmetric functions arise as matrix elements of Lie algebra elements in the subspace of symmetric tensors of a particular level-0 module which can be identified with the small quantum cohomology ring of the $k$-fold product of projective space. The analogous construction in the subspace of alternating tensors gives the known set of cylindric Schur functions which are related to the small quantum cohomology ring of Grassmannians. We prove that cylindric Schur functions form a subcoalgebra in $\Lambda$ whose structure constants are the 3-point genus 0 Gromov-Witten invariants. We show that the new families of cylindric functions obtained from the subspace of symmetric tensors also share the structure constants of a symmetric Frobenius algebra, which we define in terms of tensor multiplicities of the generalised symmetric group $G(n,1,k)$., Comment: 63 pages, 5 figures (v3: version accepted for publication in Algebraic Combinatorics)

In this paper, we consider an integral basis for affine vertex algebra Vk (sl2) when the level k is integral by a direct calculation, then use the similar way to analyze an integral basis for Virasoro vertex algebra Vvir (2k,0). Finally, we take the combination of affine algebras and Virasoro Lie algebras into consideration. By analogy with the construction of Lie algebras over Z using Chevalley bases, we utilize the Z-basis of Lav whose structure constants are integral to find an integral basis for the universal enveloping algebra of it.

Knutson, Tao, and Woodward formulated a Littlewood-Richardson rule for the cohomology ring of Grassmannians in terms of puzzles. Vakil and Wheeler-Zinn-Justin have found additional triangular puzzle pieces that allow one to express structure constants for $K$-theory of Grassmannians. Here we introduce two other puzzle pieces of hexagonal shape, each of which gives a Littlewood-Richardson rule for $K$-homology of Grassmannians. We also explore the corresponding eight versions of $K$-theoretic Littlewood-Richardson tableaux., 18 pages, 8 figures

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the this paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch–Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D cases, followed by an application to the computation of the unique normal form of the Rössler equation.

We propose a Leibniz algebra, to be called DD$^+$, which is a generalization of the Drinfel'd double. We find that there is a one-to-one correspondence between a DD$^+$ and a Jacobi--Lie bialgebra, extending the known correspondence between a Lie bialgebra and a Drinfel'd double. We then construct generalized frame fields $E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+$ satisfying the algebra $\mathcal{L}_{E_A}E_B = - X_{AB}{}^C\,E_C\,$, where $X_{AB}{}^C$ are the structure constants of the DD$^+$ and $\mathcal{L}$ is the generalized Lie derivative in double field theory. Using the generalized frame fields, we propose the Jacobi-Lie T-plurality and show that it is a symmetry of double field theory. We present several examples of the Jacobi-Lie T-plurality with or without Ramond-Ramond fields and the spectator fields., 30 pages; v2: minor corrections, references added; v3 matches the published version: extended the discussion, an appendix and references added

Squeezing of quantum states is of great interest due to its application in high-precision measurement that can exceed standard quantum noise limit described by Heisenberg uncertainty relations. Here, we study the squeezing of quantum systems with $\text{SU}(3)$ symmetry via a one-axis twisting Hamiltonian and examine the intriguing connections between the squeezing of spin-1/2 and spin-1 systems. There are seven subalgebras of $\text{su}(3)$ Lie algebra. All these subalgebras are identical with $\text{su}(2)$ Lie algebra but not all of them have the same anticommutation relations as $\text{su}(2)$. Interestingly, squeezing parameters corresponding to spin-1 subalgebras depend not only on structure constants but also on anticommutation relations of the subalgebras. Our results are reported for the subalgebras with vanishing anticommutators and nematic squeezing, while in other cases our first-principle calculation recovers known results.

We describe the effect of the marginal deformation of the $\cal N = (4, 4)$ superconformal $(T^4)^N /S_N$ orbifold theory on a doublet of R-neutral twisted Ramond fields, in the large-$N$ approximation. Our analysis of their dynamics explores the explicit analytic form of the genus-zero four-point function involving two R-neutral Ramond fields and two deformation operators. We compute this correlation function with two different approaches: the Lunin-Mathur path-integral technique and the stress-tensor method. From its short distance limits, we extract the OPE structure constants and the scaling dimensions of non-BPS fields appearing in the fusion. In the deformed CFT, at second order in the deformation parameter, the two-point function of the $n$-twisted Ramond fields is UV-divergent. We perform an appropriate regularization, together with a renormalization of the undeformed fields, obtaining finite, well-defined corrections to their two-point functions and their bare conformal weights, for $n < N$. The fields with maximal twist $n=N$ remain protected from renormalization, with vanishing anomalous dimensions., 27 pages, 1 figure; V2 - corrected and improved version; V3 - 37 pages, 1 figure; extended, improved version; new discussions and references added; to appear in JHEP