Vector-Valued Polynomials and a Matrix Weight Function with B2-Action

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on R N . The Dunkl operators are denoted by T j for 1 ≤ j ≤ N , and the Laplacian Δ κ = ∑ j = 1 N T j 2 . This paper finds the homogeneous harmonic polynomials annihilated by all T j for j > 2 . The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.

For each partition τ of N, there are irreducible modules of the symmetric groups S N and of the corresponding Hecke algebra H N t whose bases consist of the reverse standard Young tableaux of shape τ . There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials constructed by Griffeth for the infinite family G n , p , N of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For each of the groups S N and the Hecke algebra H N t , there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by κ and q , t , respectively. For certain values of these parameters (called singular values), there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is x 1 m ⊗ S , where S is an arbitrary reverse standard Young tableau of shape τ . The singular values depend on the properties of the edge of the Ferrers diagram of τ .

In this paper, we mainly discuss linear deformations of a Hom-Lie-Yamaguti algebra and introduce the notion of a Hom-Nijenhuis operator. We introduce the notion of a product structure on a Hom-Lie-Yamaguti algebra, which is a Hom-Nijenhuis operator E satisfying E2 = Id. There is a product structure on an involutive Hom-Lie-Yamaguti algebra if and only if the Hom-Lie-Yamaguti algebra is the direct sum of two subalgebras (as vector spaces). At the same time, we also introduce the notion of a complex structure on a Hom-Lie-Yamaguti algebra. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on an involutive Hom-Lie-Yamaguti algebra.

There is an algebra of commutative differential-difference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators\(T_i : = \frac{\partial }{{\partial x_i }} + k\sum\nolimits_{j \ne i} {\frac{{1 - (ij)}}{{x_i - x_j }}} \), (i=1, ...,N, where (ij) denotes the transposition of the variablesxixj andk is a fixed parameter). We introduce a family of functions {pα}, indexed bym-tuples of non-negative integers α = (α1, ..., αm) form≤N, which allow a workable treatment of important constructions such as the intertwining operatorV. This is a linear map on polynomials, preserving the degree of homogeneity, for which\(T_i V = V\frac{\partial }{{\partial x_i }}\),i = 1, ...,N, normalized byV1=1 (seeDunkl, Canadian J. Math.43 (1991), 1213–1227). We show thatTipα=0 fori>m, and $$V(x_1^{\alpha _1 } \cdots x_m^{\alpha _m } ) = \frac{{\lambda _1 !\lambda _2 ! \cdots \lambda _m !}}{{\left( {Nk + 1} \right)_{\lambda _1 } \left( {Nk - k + 1} \right)_{\lambda _2 } \cdots (Nk - (m - 1)k + 1)_{\lambda _m } }}p_\alpha + \sum\limits_\beta {A_{\beta \alpha } p_{\beta ,} } $$ where (λ1, λ2, ..., λm) is the partition whose parts are the entries of α (That is, λ1➮ λ2➮ ... λm➮0), β = (β1, ..., βm), ∑i=1m βi = ∑i=1m αm and the sorting of β is a partition strictly larger than λ in the dominance order. This triangular matrix representation ofV allows a detailed study. There is an inner product structure on span {pα} and a convenient set of self-adjoint operators, namelyTiρi, whereρipα ≔p(α1, ...., αi + 1, ..., αm). This structure has a bi-orthogonal relationship with the Jack polynomials inm variables. Values ofk for whichV fails to exist are called singular values and were studied byDe Jeu, Opdam, andDunkl in Trans. Amer. Math. Soc.346 (1994), 237–256. As a partial verification of a conjecture made in that paper, we construct, for anya=1,2,3,... such that gcd(N−m+1,a)<(N−m+1)/m andm≤N/2, a space of polynomials annihilated by eachTi fork=−a/(N−m+1) and on which the symmetric groupSN acts according to the representation (N−m, m).

Let $\mathbb{F}_q$ be an arbitrary finite field, and $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Let $Δ(\mathcal{E})$ be the set of distances determined by pairs of points in $\mathcal{E}$. By using the Kloosterman sums, Iosevich and Rudnev proved that if $|\mathcal{E}|\ge 4q^{\frac{d+1}{2}}$, then $Δ(\mathcal{E})=\mathbb{F}_q$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-plane incidence bound due to Rudnev to prove that if $\mathcal{E}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $(d+1)/2$, and still cover all distances. We also show that the number of pairs of points in $\mathcal{E}$ of any given distance is close to its expected value.

We build an irreducible unitary representation of SO( ∞ ) from the usual one of SO( n ) in the space of harmonic homogeneous polynomials of degree m of ℝn. We give a characterization of these new representations which extends in a natural way the finite dimensional characterization. In the particular case of SO( ∞ ), we thus get some results of Ol‘shanskii (cf. [12]). This leads to a new proof of McKean conjecture about irreducible representations of ( infin ) (cf. [10]).

Let V be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH*(OG) and show that its product structure is determined by the ring of -polynomials. A ‘quantum Schubert calculus’ is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov--Witten invariants. As an application, we show that the table of three-point, genus zero Gromov–Witten invariants for OG coincides with that for a corresponding Lagrangian Grassmannian LG, up to an involution.

The main object of this chapter is to construct and study the irreducible polynomial representations of the general linear group GL m ℂ = GL ( E ), where E is a complex vector space of dimension m . These can be formed by a basic construction in linear algebra that generalizes a well known construction of symmetric and exterior products; they make sense for any module over a commutative ring. These representations are parametrized by Young diagrams λ with at most m rows, and have bases corresponding to Young tableaux on λ with entries from [ m ]. They can also be constructed from representations of symmetric groups. Like the latter, these have useful realizations both as subspaces and as quotient spaces of naturally occurring representations, with relations given by quadratic equations. The characters of the representations are given in §8.3. To prove that these give all the irreducible representations we use a bit of the Lie group–Lie algebra story, which is sketched in this setting in §8.2. In the last section we describe some variations on the quadratic equations. In particular, we identify the sum of all polynomial representations with a ring constructed by Deruyts a century ago. A construction in linear algebra For any commutative ring R and any R -module E , and any partition λ, we will construct an R -module denoted E λ . (For applications in these notes, the case where R = ℂ, so E is a complex vector space, will suffice.)

Following a recent proposal of Richard Borcherds to regard fusion as the ringlike tensor product of modules of a quantum ring, a generalization of rings and vertex algebras, we define fusion as a certain quotient of the (vector space) tensor product of representations of the symmetry algebra [Formula: see text]. We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of [Formula: see text] on it, under which the central extension is preserved. Having defined fusion in this way, determining the fusion rules is then the algebraic problem of decomposing the tensor product into irreducible representations. We demonstrate how to solve this for the case of the WZW and the minimal models and recover thereby the well-known fusion rules. The action of the symmetry algebra on the tensor product is given in terms of a comultiplication. We calculate the R matrix of this comultiplication and find that it is triangular. This seems to shed some new light on the possible rôle of the quantum group in conformal field theory.

1. Let R be the additive group of all real numbers and t → Tt a representation of this group on an n-dimensional complex vector space X. This means that to each real number t a linear operator Tt : X → X is assigned in such a way that $${\text{T}}_{{\text{t + s}}} = {\text{T}}_{\text{t}} {\text{T}}_{\text{s}} , {\text{T}}_0 = {\text{I}}$$ (1) Since the set {Tt : t ∈ R} consists of operators which commute one to the other the irreducible representations are one dimensional. Hence we are lead to find all functions f: R → C (C is the set of all complex numbers) such that $${\text{f}}\left( {{\text{t}}\,{\text{ + }}\,{\text{s}}} \right) = {\text{f}}\left( {\text{t}} \right)\,\,{\text{f}}\,\,{\text{(s),}}\,\,\,\,\,{\text{f}}\,\,{\text{(0)}}\,{\text{ = }}\,\,{\text{1}}{\text{.}}$$ (2)

Let $G_M$ be one of the complex Lie groups $O_M$ and $Sp_M$. The irreducible finite-dimensional representations of the group $G_M$ are labeled by partitions $\mu$ satisfying certain extra conditions. Let $U$ be the representation of $G_M$ corresponding to $\mu$. Regard the direct product $G_N\times G_M$ as a subgroup of $G_{N+M}$. Let $V$ be the irreducible representation of $G_{N+M}$ corresponding to a partition $\lambda$. Consider the vector space $W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $G_N$. Let $n$ be sum of parts of $\lambda$ less the sum of parts of $\mu$. For any choice of a standard Young tableau of skew shape $\lambda/\mu$, we realize $W$ as a subspace in the tensor product of $n$ copies of the defining $N$-dimensional representation of $G_N$. This subspace is determined as the image of a certain linear operator $F(M)$ in the tensor product, given by an explicit formula. When M=0 and $W=V$ is an irreducible representation of $G_N$, we recover the classical realization of $V$ as a subspace in the space of all traceless tensors. Then the operator F(0) can be regarded as the analogue for $G_N$ of the Young symmetrizer, corresponding to the chosen standard tableau of shape $\lambda$. Even in the special case M=0, our formula for the operator $F(M)$ is new. Our results are applications of representation theory of the twisted Yangian, corresponding to $G_N$. In particular, $F(M)$ is an intertwining operator between two representations of the twisted Yangian in the $n$-fold tensor product.

Our object of study is the ring of invariant differential operators on a hermitian symmetric space G/K of classical and noncompact type. Here, as usual, G is a connected noncompact semisimple Lie group with finite center and K is a maximal compact subgroup of G. It is well-known that the ring, denoted by ?(G/K), is isomorphic to a polynomial ring of 1 variables, 1 being the rank of G/K. This is true even in the nonhermitian case. If 1 = 1, it is generated by the Laplace-Beltrami operator z/ which is essentially self-adjoint; moreover -Iz2 is nonnegative. In the general case, an easily posable problem is to find an explicitly defined set of generators. We can go one step further by focusing our attention on the nonnegativity, which necessarily limits the range of eigenvalues under a suitable integrability condition on functions. It is thus natural to ask whether there exist some canonically defined operators which generate -9(G/K) and have the property of nonnegativity. The main purpose of the present paper is to give an affirmative answer to this question. In fact, our answer applies to a somewhat more general type of rings of differential operators which includes 9(G/K) as a special case. To be explicit, we take an irreducible representation p: K -* GL(V) with a complex vector space V of finite dimension, and consider the set C'(p) of all V-valued Cx functions f on G such that f(xk -1) = p(k)f(x) for every k e K. We then denote by ?(p) the ring of left-invariant difrrential operators on G which map C'(p) into itself. Now the complexification g of the Lie algebra of G has abelian subalgebras p+ and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on G/K at the origin. For any complex vector space W let Sr(W) denote the ring of all complex-valued homogeneous polynomial functions on W of degree r, and let S(W) = Er=0Sr(W). Through the adjoint representation of G on g, K acts naturally on Sr( ?+). It is a known fact, due to Hua and Schmid, that each irreducible constituent of this representation of K has multiplicity one. Now our principal

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a “free” polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.

This paper concerns a faithful representation [math] of a simple linear algebraic group [math] . Under mild assumptions, we show that if [math] is large enough, then the Lie algebra of [math] acts generically freely on [math] . That is, the stabilizer in [math] of a generic vector in [math] is zero. The bound on [math] grows like [math] and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position.

‎Let $V$ be a unitary space‎. ‎Suppose $G$ is a subgroup of the symmetric group of degree $m$ and $\Lambda$ is an irreducible unitary representation of $G$ over a vector space $U$‎. ‎Consider the generalized symmetrizer on the tensor space $U\otimes V^{\otimes m}$‎, ‎$$ S_{\Lambda}(u\otimes v^{\otimes})=\dfrac{1}{|G|}\sum_{\sigma\in G}\Lambda(\sigma)u\otimes v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}(m)} $$ defined by $G$ and $\Lambda$‎. ‎The image of $U\otimes V^{\otimes m}$ under the map $S_\Lambda$ is called the generalized symmetry class of tensors associated with $G$ and $\Lambda$ and is denoted by $V_\Lambda(G)$‎. ‎The elements in $V_\Lambda(G)$ of the form $S_{\Lambda}(u\otimes v^{\otimes})$ are called generalized decomposable tensors and are denoted by $u\circledast v^{\circledast}$‎. ‎For any linear operator $T$ acting on $V$‎, ‎there is a unique induced operator $K_{\Lambda}(T)$ acting on $V_{\Lambda}(G)$ satisfying $$ K_{\Lambda}(T)(u\otimes v^{\otimes})=u\circledast Tv_{1}\circledast \cdots \circledast Tv_{m}‎. ‎$$ If $\dim U=1$‎, ‎then $K_{\Lambda}(T)$ reduces to $K_{\lambda}(T)$‎, ‎induced operator on symmetry class of tensors $V_{\lambda}(G)$‎. ‎In this paper‎, ‎the basic properties of the induced operator $K_{\Lambda}(T)$ are studied‎. ‎Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions‎.