Type A Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Positive Integer ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Algebraic Number ◽  
Type A ◽  
Algebraic Number Field ◽  
Basis Vector ◽  
Roots Of Unity ◽  
Finite Set ◽  
Multiple Dirichlet Series

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are introduced: Φ‎, a reduced root system; n, a positive integer; F, an algebraic number field containing the group μ‎₂ₙ of 2n-th roots of unity; S, a finite set of places of F containing all the archimedean places, all places ramified over a ℚ; and an r-tuple of nonzero S-integers. In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern can be read from differences of consecutive row sums in the pattern. The chapter considers in this case expressions of the weight of the pattern up to an affine linear transformation.

Statement B and the Yang-Baxter Equation

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Statistical Mechanics ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Transfer Matrix ◽  
Baxter Equation ◽  
Partition Functions ◽  
Alternate Proof ◽  
Crystal Basis ◽  
Exactly Solved Models ◽  
Multiple Dirichlet Series

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

The hyperelliptic equation over function fields

1983 ◽  
Vol 93 (2) ◽  
pp. 219-230 ◽  
Author(s):  
R. C. Mason
Keyword(s):  
Elliptic Equation ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Integer Coefficients ◽  
Finite Set ◽  
Integer Solutions ◽  
Explicit Bounds

Siegel, in a letter to Mordell of 1925(9), proved that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree at least three with integer coefficients. Siegel's method reduces the hyperelliptic equation to a finite set of Thue equations f(x, y) = 1, where f denotes a binary form with algebraic coefficients and at least three distinct linear factors; x and y are integral in a fixed algebraic number field. Siegel had already proved that the Thue equations so obtained have only finitely many solutions. However, as is well known, the work of Siegel is ineffective in that it fails to provide bounds on the integer solutions of y2 = g(x). In 1969 Baker (1), using the theory of linear forms in logarithms, employed Siegel's technique to establish explicit bounds on x and y; Baker's result thus reduced the problem of determining all integer solutions of the hyperelliptic equation to a finite amount of computation.

Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Riemann Zeta Function ◽  
Functional Equations ◽  
Lattice Points ◽  
Baxter Equation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Multiple Dirichlet Series

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.

Simultaneous rational approximations to certain algebraic numbers

1967 ◽  
Vol 63 (3) ◽  
pp. 693-702 ◽  
Author(s):  
A. Baker
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Infinitely Many Solutions ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Famous Theorem ◽  
Image Position

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

scholarly journals On the p-parts of quadratic Weyl group multiple Dirichlet series

2008 ◽  
Vol 2008 (623) ◽  
pp. 1-23 ◽  
Author(s):  
Gautam Chinta ◽  
Solomon Friedberg ◽  
Paul E. Gunnells
Keyword(s):  
Dirichlet Series ◽  
Weyl Group ◽  
Multiple Dirichlet Series

Crystals and Gelfand-Tsetlin Patterns

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Dirichlet Series ◽  
Weyl Group ◽  
Crystal Bases ◽  
Cartan Type ◽  
Weight Lattice ◽  
Crystal Graphs ◽  
Multiple Dirichlet Series

This chapter translates the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and circling rules in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated functions on Gelfand-Tsetlin patterns in the previous chapter now takes on intrinsic representation theoretic meaning. The discussion is restricted to crystals of Cartan type Aᵣ. The Weyl vector, denoted by ρ‎, is considered as an element of the weight lattice, and the bijection between Gelfand-Tsetlin patterns and tableaux is described. The chapter also examines the λ‎-part of the multiple Dirichlet series in terms of crystal graphs.

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