Affine root systems and Dedekind's?-function

The Structure of Total Reflection Orders in Affine Root Systems

Reflectable bases for affine reflection systems

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

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.

The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type $A$ Weyl denominator formula and the use of a polynomial argument. In this paper, we rephrase the renormalized Weyl-Kac denominator formula as a sum parametrized by affine Grassmannian elements. This naturally gives rise to the (dual) atomic length of the root system considered introduced by Chapelier-Laget and Gerber. We then provide an interpretation of this atomic length as the cardinality of some subsets of $n$-core partitions by using foldings of affine Dynkin diagrams. This interpretation does not permit the direct use of a polynomial argument for all affine root systems. We show that this obstruction can be overcome by computing the atomic length of certain families of integer partitions. Then we show how hook-length statistics on these partitions are connected with the Coxeter length on affine Grassmannian elements and Nekrasov-Okounkov type formulas.

A formal identity for affine root systems

We study the non-perturbative dynamics of the two dimensional O(N ) and Grassmannian sigma models by using compactification with twisted boundary conditions on $$ \mathbb{R}\times {S}^1 $$ , semi-classical techniques and resurgence. While the O(N) model has no instantons for N > 3, it has (non-instanton) saddles on $$ {\mathbb{R}}^2 $$ , which we call 2d-saddles. On $$ \mathbb{R}\times {S}^1 $$ , the resurgent relation between perturbation theory and non-perturbative physics is encoded in new saddles, which are associated with the affine root system of the o(N ) algebra. These events may be viewed as fractionalizations of the 2d-saddles. The first beta function coefficient, given by the dual Coxeter number, can then be intepreted as the sum of the multiplicities (dual Kac labels) of these fractionalized objects. Surprisingly, the new saddles in O(N ) models in compactified space are in one-to-one correspondence with monopole-instanton saddles in SO(N ) gauge theory on $$ {\mathbb{R}}^3\times {S}^1 $$ . The Grassmannian sigma models Gr(N, M ) have 2d instantons, which fractionalize into N kink-instantons. The small circle dynamics of both sigma models can be described as a dilute gas of the one-events and two-events, bions. One-events are the leading source of a variety of non-perturbative effects, and produce the strong scale of the 2d theory in the compactified theory. We show that in both types of sigma models the neutral bion emulates the role of IR-renormalons. We also study the topological theta angle dependence in both the O(3) model and Gr(N, M ), and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes, providing a microscopic argument for Haldane’s conjecture.

We introduce an elliptic extension of Dyson’s Brownian motion model, which is a temporally inhomogeneous diffusion process of noncolliding particles defined on a circle. Using elliptic determinant evaluations related to the reduced affine root system of type \(A\), we give determinantal martingale representation (DMR) for the process, when it is started at the configuration with equidistant spacing on the circle. DMR proves that the process is determinantal and the spatio-temporal correlation kernel is determined. By taking temporally homogeneous limits of the present elliptic determinantal process, trigonometric and hyperbolic versions of noncolliding diffusion processes are studied.

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n > 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.