Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements

Previous article Next article Sign Variations of the Macdonald IdentitiesDennis StantonDennis Stantonhttps://doi.org/10.1137/0517103PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractSign variations are given for the Macdonald identities for root systems of small rank. Limiting cases of these identities give properties of the eta function. Two such formulas are explicitly given.[1] Richard Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal., 11 (1980), 938–951 10.1137/0511084 82e:33002 0458.33002 LinkISIGoogle Scholar[2] F. Dyson, 1968, letter to L. Winquist, March Google Scholar[3] I. Macdonald, Affine root systems and Dedekind's $\eta$-function, Invent. Math., 15 (1972), 91–143 50:9996 0244.17005 CrossrefISIGoogle Scholar[4] I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal., 13 (1982), 988–1007 10.1137/0513070 84h:17006a 0498.17006 LinkISIGoogle Scholar[5] S. Milne, An elementary proof of the Macdonald identities for $A\sp {(1)}\sb l$, Adv. in Math., 57 (1985), 34–70 87e:17020 0586.33011 CrossrefISIGoogle Scholar[6] Lasse Winquist, An elementary proof of $p(11m+6)\equiv 0\,({\rm mod}\ 11)$, J. Combinatorial Theory, 6 (1969), 56–59 38:4434 0241.05006 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Leading coefficients of Morris type constant term identitiesAdvances in Applied Mathematics, Vol. 87 Cross Ref Logarithmic and Complex Constant Term Identities29 July 2013 Cross Ref A q-product Tutorial for a q-series Maple Package Cross Ref A New Proof of Winquist's IdentityJournal of Combinatorial Theory, Series A, Vol. 78, No. 2 Cross Ref Macdonald’s constant term conjectures for exceptional root systems1 January 1991 | Bulletin of the American Mathematical Society, Vol. 24, No. 2 Cross Ref A Stembridge-Stanton style elementary proof of the Habsieger-Kadell q-Morris identityDiscrete Mathematics, Vol. 79, No. 3 Cross Ref The crank of partitions mod 8,9 and 101 January 1990 | Transactions of the American Mathematical Society, Vol. 322, No. 1 Cross Ref An Elementary Approach to the Macdonald Identities Cross Ref A Unified Approach to Macdonald’s Root-System ConjecturesDoron Zeilberger17 July 2006 | SIAM Journal on Mathematical Analysis, Vol. 19, No. 4AbstractPDF (2886 KB)First layer formulas for characters of 𝑆𝐿(𝑛,𝐶)1 January 1987 | Transactions of the American Mathematical Society, Vol. 299, No. 1 Cross Ref Volume 17, Issue 6| 1986SIAM Journal on Mathematical Analysis History Submitted:17 December 1984Accepted:16 September 1985Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0517103Article page range:pp. 1454-1460ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

We introduce a series of conjectured identities that deform Weyl's denominator formula and generalize Tokuyama's formula to other root systems. These conjectures generalize a number of well-known results due to Okada. We also prove a related result for $B'_n$ that generalizes a theorem of Simpson. Nous proposons une série de conjectures qui sont des déformations de la formule dénominateur de Weyl et qui généralisent la formule de Tokuyama à d’autres systèmes de racines. Ces résultats sont des généralisations de théorèmes bien connus dus à Okada. Nous donnons aussi la preuve d’un résultat pour $B'_n$ qui est une généralisation d’un théorème de Simpson.

Reflectable bases for affine reflection systems

In 2010, G.-N. Han obtained the generating function for the number of size t t hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even t t . If n t ( λ ) n_t(\lambda ) is the number of size t t hooks in a partition λ \lambda and S C \mathcal {SC} denotes the set of self-conjugate partitions, then for even t t we have ∑ λ ∈ S C x n t ( λ ) q | λ | = ( − q ; q 2 ) ∞ ⋅ ( ( 1 − x 2 ) q 2 t ; q 2 t ) ∞ t 2 . \begin{equation*} \sum _{\lambda \in \mathcal {SC}} x^{n_t(\lambda )} q^{\vert \lambda \vert } = (-q;q^2)_{\infty } \cdot ((1-x^2)q^{2t};q^{2t})_{\infty }^{\frac {t}{2}}. \end{equation*} As a consequence, if a t ⋆ ( n ) a_t^{\star }(n) is the number of such hooks among the self-conjugate partitions of n n , then for even t t we obtain the simple formula a t ⋆ ( n ) = t ∑ j ≥ 1 q ⋆ ( n − 2 t j ) , \begin{equation*} a_t^{\star }(n)=t\sum _{j\geq 1} q^{\star }(n-2tj), \end{equation*} where q ⋆ ( m ) q^{\star }(m) is the number of partitions of m m into distinct odd parts. As a corollary, we find that t ∣ a t ⋆ ( n ) t\mid a_t^{\star }(n) , which confirms a conjecture of Ballantine, Burson, Craig, Folsom and Wen.

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of t t -hooks in the partitions of n n . We prove that the limiting distribution is normal with mean \[ μ t ( n ) ∼ 6 n π − t 2 \mu _t(n)\sim \frac {\sqrt {6n}}{\pi }-\frac {t}{2} \] and variance \[ σ t 2 ( n ) ∼ ( π 2 − 6 ) 6 n 2 π 3 . \sigma _t^2(n)\sim \frac {(\pi ^2-6)\sqrt {6n}}{2\pi ^3}. \] Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed t ≥ 4 t\geq 4 in partitions of n n converge to a shifted Gamma distribution with parameter k = ( t − 1 ) / 2 k=(t-1)/2 and scale θ = 2 / ( t − 1 ) \theta =\sqrt {2/(t-1)} .

The fact $$\lim _{p \rightarrow 0} \theta (\zeta ; p)=1-\zeta $$ , $$\zeta \in \mathbb {C}^{\times }$$ suggests that the theta function $$\theta (\zeta ; p)$$ is an elliptic analogue of a linear function of $$\zeta $$ . What is the elliptic analogue of a polynomial of $$\zeta $$ ? Rosengren and Schlosser gave seven kinds of answers to this fundamental question by introducing seven infinite series of spaces of theta functions associated with the irreducible reduced affine root systems, $$R_n=A_{n-1}$$ , $$B_n$$ , $$B^{\vee }_n$$ , $$C_n$$ , $$C^{\vee }_n$$ , $$BC_n$$ , $$n \in \mathbb {N}$$ , and $$D_n$$ , $$n \in \{2, 3, \dots \}$$ . Here n indicates the degree of the elliptic analogues of polynomials. The basis functions for the function spaces are called the $$R_n$$ theta functions and are denoted by $$\{\psi ^{R_n}_j\}_{j=1}^n$$ . It was proved that the determinants consisting of $$\{\psi ^{R_n}_j\}_{j=1}^n$$ provide the Macdonald denominator formulas, which are the elliptic extensions of the Weyl denominator formulas. In this chapter, first we give a brief review of the theory of Rosengren and Schlosser. Then we introduce appropriate inner products and prove the biorthogonality relations for the $$R_n$$ theta functions of Rosengren and Schlosser.

The Structure of Total Reflection Orders in Affine Root Systems

A root system is any collection of vectors that has properties that satisfy the roots of a semi simple Lie algebra. If g is semi simple, then the root system A, (Q) can be described as a system of vectors in a Euclidean vector space that possesses some remarkable symmetries and completely defines the Lie algebra of g. The purpose of this paper is to show the essentiality of the root system on the Lie algebra. In addition, the paper will mention the connection between the root system and Ways chambers. In addition, we will show Dynkin diagrams, which are an integral part of the root system.

Root systems and inflations of non-negative quasi-Cartan matrices

Affine root systems and Dedekind's?-function

Extending π-systems to bases of root systems

The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the $A_N$ affine root system, enumerated according to the cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory, where the equations of motion are invariant with respect to permutations of all discrete coordinates. The discrete evolution operator is constructed explicitly. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit $L,N\to\infty$. Some conjectures about the structure of the spectrum of the corresponding discrete models are stated.

Given the toric (or toral) arrangement defined by a root system \Phi , we classify and count its components of each dimension. We show how to reduce to the case of 0-dimensional components, and in this case we give an explicit formula involving the maximal subdiagrams of the affine Dynkin diagram of \Phi . Then we compute the Euler characteristic and the Poincar\'{e} polynomial of the complement of the arrangement, that is the set of regular points of the torus.

A formal identity for affine root systems

Our aim in this chapter is to study the classical Lie algebras sl(n, C), so(n, C), and sp(n, C) for n ≥ 2. We shall show that, with two exceptions, all these Lie algebras are simple. We shall also find their root systems and the associated Dynkin diagrams and describe their Killing forms. The main result we prove is the following theorem.