Symmetric polynomials and 𝑈_{𝑞}(̂𝑠𝑙₂)

A general scheme of construction of Drinfeldians and Yangians from quantum non-twisted affine Kac-Moody algebras is presented. Explicit description of Drinfeldians and Yangians for all Lie algebras of the classical series A, B, C, D is given in terms of a Chevalley basis. Rational and trigonometric deformations of a universal enveloping algebra U(g(u)) (g is a finite-dimensional complex simple Lie algebra) - Yangian Yη(g) and a quantum (q-deformed) affine algebraUq(g(u)) - are playing increasingly a role in the theory of integrable systems and the quantum field theory. For applications it is useful to know various realizations (or bases) of these deformations. Among various realizations the Chevalley basis plays a special role due to its compactness and simplicity. In the case of the quantum affine algebras the Chevalley basis was originally introduced in the defining relations while the Yangians were initially de- fined in terms of a finite Cartesian basis (the first Drinfeld realization (2)) as well as in terms of a special infinite basis (the second Drinfeld realization (3)). It was not clear whether does exist a Chevalley basis for the Yangians. Recently in (12, 13) it was shown that the Yangian can be realized in terms of Chevalley basis and, moreover, there was found a simple connection of this realization with the corresponding quantum affine algebra (also see (14, 15)). This connection allows to introduce a new Hopf algebra called Drinfeldian which depends on two deformation parameters q and η. The Drinfeldian Dqη(g) can be considered as quantization of U(g(u)) in the direction of a classical r-matrix which is a sum of simplest rational and trigonometric r-matrices. When the parameter q goes to 1 we obtain the Yangian Yη(g) in terms of the Chevalley basis. In this paper we remind once more the procedure of construction of the Drin- feldian Dqη(g) and the Yangian Yη(g) in terms of the Chevalley basis starting from

Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood---Richardson coefficient for a triple of Schur polynomials is 1, then the corresponding coefficient for Jack polynomials can be expressed as a product of weighted hooks of the Young diagrams associated with the partitions indexing the coefficient. We prove a special case of this conjecture in which the partitions indexing the Littlewood---Richardson coefficient have at most three parts. We also show that this result extends to Macdonald polynomials.

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schützenberger) could be used to define a new basis for the ring of quasisymmetric functions we call “Quasisymmetric Schur functions” (QS functions for short). In this paper we develop the combinatorics of these polynomials further, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coefficients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis.

The skew Schubert polynomials are those that are indexed by skew elements of the Weyl group, in the sense of Tamvakis [J. reine angew. Math. 652 (2011), 207–244]. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where the tableaux used are fillings of the associated skew Young diagram. These are the first such theorems for symplectic and orthogonal Schubert polynomials, even in the single case. We also deduce tableau formulas for double Schur, double theta, and double eta polynomials, in their specializations as double Grassmannian Schubert polynomials. The latter results generalize the tableau formulas for symmetric (and single) Schubert polynomials due to Littlewood (in type A) and the author (in types B, C, and D).

We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group. Nous utilisons des algèbres de Hopf pour prouver une version de la règle de Littlewood―Richardson pour les fonctions de Schur gauches, qui implique une conjecture d'Assaf et McNamara. Nous établissons également des règles de Littlewood―Richardson gauches pour les $P-$ et $Q-$fonctions de Schur et les fonctions de Schur rubbans non commutatives, ainsi que des règles de Pieri gauches pour les $k-$fonctions de Schur, les $k-$fonctions de Schur duales, et pour l'homologie de la Grassmannienne affine du groupe symplectique.

We present an explicit formula for the transition matrix $\\mathcal{C}$ from\nthe type $C_n$ degeneration of the Koornwinder polynomials\n$P_{(1^r)}(x\\,|\\,a,-a,c,-c\\,|\\,q,t)$ with one column diagrams, to the type\n$C_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the\nmatrix $\\mathcal{C}$ enjoy a set of three term recursion relations, which can\nbe regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or\nballot numbers. Some transition matrices are studied associated with the type\n$(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}_{(1^r)}(x\\,|\\,b;q,t)=\nP_{(1^r)}\\big(x\\,|\\,b^{1/2},-b^{1/2},q^{1/2}b^{1/2},-q^{1/2}b^{1/2}\\,|\\,q,t\\big)$.\nIt is also shown that the $q$-ballot numbers appear as the Kostka polynomials,\nnamely in the transition matrix from the Schur polynomials\n$P^{(C_n,C_n)}_{(1^r)}(x\\,|\\,q;q,q)$ to the Hall-Littlewood polynomials\n$P^{(C_n,C_n)}_{(1^r)}(x\\,|\\,t;0,t)$.\n

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.

Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by Koike and Terada and satisfy the general branching rules. Furthermore, we derive the Jacobi-Trudi identities and Gelfand-Tsetlin patterns for these symmetric functions. Additionally, the vertex operator method yields their Cauchy-type identities. This demonstrates that vertex operator representations serve not only as a tool for studying symmetric functions but also offers unified realizations for skew Schur functions of types A, C, and D.

The Jack polynomials J (x; ) form a remarkable class of symmetric polynomials. They are indexed by a partition and depend on a parameter . One of their properties is that several classical families of symmetric functions can be obtained by specializing , e.g., the monomial symmetric functions m ( =∞), the elementary functions e ′ ( = 0), the Schur functions s ( = 1) and nally the two classes of zonal polynomials ( = 2, = 1=2). The Jack polynomials can be de ned in various ways, e.g.: a) as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b) as simultaneous eigenfunctions of certain di erential operators (the Sekiguchi–Debiard operators). Recently Opdam, [O], constructed a similar family F (x; ) of nonsymmetric polynomials. The index runs now through all compositions ∈ Nn. They are de ned in a completely similar fashion, e.g., the Sekiguchi–Debiard operators are being replaced by the Cherednik di erential-re ection operators (see Sect. 3). It is becoming more and more clear that these polynomials are as important as their symmetric counterparts. The purpose of this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the F together with two formulas to obtain J from them. d) combinatorial formulas of both J and F in terms of certain generalized tableaux.

Weak amenability of a weighted group algebra, or a Beurling algebra, is a long-standing open problem. The commutative case has been extensively investigated and fully characterized. We study the non-commutative case. Given a weight function $\omega$ on a locally compact group $G$, we characterize derivations from $L^1(G,\omega)$ into its dual in terms of certain functions. Then we show that for a locally compact IN group $G$, if there is a non-zero continuous group homomorphism $\varphi$: $G\to \mathbb{C}$ such that $\varphi(x)/\omega(x)\omega(x^{-1})$ is bounded on $G$, then $L^1(G,\omega)$ is not weakly amenable. Some useful criteria that rule out weak amenability of $L^1(G,\omega)$ are established. Using them we show that for many polynomial type weights the weighted Heisenberg group algebra is not weakly amenable, neither is the weighted $\boldsymbol{ax+b}$ group algebra. We further study weighted quotient group algebra $L^1(G/H,\hat\omega)$, where $\hat\omega$ is the canonical weight on $G/H$ induced by $\omega$. We reveal that the kernel of the canonical homomorphism from $L^1(G,\omega)$ to $L^1(G/H,\hat\omega)$ is complemented. This allows us to obtain some sufficient conditions under which $L^1(G/H,\hat\omega)$ inherits weak amenability of $L^1(G,\omega)$. We study further weak amenability of Beurling algebras of subgroups. In general, weak amenability of a Beurling algebra does not pass to the Beurling algebra of a subgroup. However, in some circumstances this inheritance can happen. We also give an example to show that weak amenability of both $L^1(H,\omega|_H)$ and $L^1(G/H,\hat\omega)$ does not ensure weak amenability of $L^1(G,\omega)$.

The Schur polynomials s λ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For ρ=(n,n-1,⋯,1) a staircase shape and μ⊆ρ a subpartition, the Stembridge equality states that s ρ/μ =s ρ/μ T . This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials G λ , and the dual stable Grothendieck polynomials g λ , developed by Buch, Lam, and Pylyavskyy, are variants of the Schur polynomials and describe the K-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood–Richardson rule, we prove that G ρ/μ =G ρ/μ T and g ρ/μ =g ρ/μ T , the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials.

Absolute values of L-functions for [formula omitted] at the point 1

The notion of the connectivity class of minimal words in the algebra Hn(q) is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented. Guided by these results, the connection between the Ocneanu trace on Hn(q) and Schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of Hn(q). They appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to Schur functions. Their evaluation involves the use of the Littlewood–Richardson rule, which is proved to apply to Hn(q) just as it does to Sn. Both representation matrices and characters are tabulated.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We study the distribution of the trace of the seven-dimensional representation of the exceptional compact simple Lie group G G of type G 2 G_{2} . The interest for this distribution comes from its relevance to the equidistribution of several families of exponential sums involving a seven degree binomial phase, established by N. Katz. Firstly, we give a construction of the algebraic group of type G 2 G_{2} and of its Lie algebra, whose G G is the compact form. We then define the Steinberg map of G G , defined from the traces of the fundamental representations, inducing a homeomorphism from the alcove of G G (the simplex parametrizing conjugacy classes) to a compact set in the affine space. By combining Weyl’s integration formula and the Steinberg map, we obtain an explicit expression for the probability density function of the distribution of the trace function on G G in terms of Gauss hypergeometric function, and some other special functions. This answers a question raised by J.-P. Serre and N. Katz.