72,488 research outputs found
Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
We show that the geometric lifting of the RSK correspondence introduced by
A.N. Kirillov (2001) is volume preserving with respect to a natural product
measure on its domain, and that the integrand in Givental's integral formula
for GL(n,R)-Whittaker functions arises naturally in this context. Apart from
providing further evidence that Whittaker functions are the natural analogue of
Schur polynomials in this setting, our results also provide a new
`combinatorial' framework for the study of random polymers. When the input
matrix consists of random inverse gamma distributed weights, the probability
distribution of a polymer partition function constructed from these weights can
be written down explicitly in terms of Whittaker functions. Next we restrict
the geometric RSK mapping to symmetric matrices and show that the volume
preserving property continues to hold. We determine the probability law of the
polymer partition function with inverse gamma weights that are constrained to
be symmetric about the main diagonal, with an additional factor on the main
diagonal. The third combinatorial mapping studied is a variant of the geometric
RSK mapping for triangular arrays, which is again showed to be volume
preserving. This leads to a formula for the probability distribution of a
polymer model whose paths are constrained to stay below the diagonal. We also
show that the analogues of the Cauchy-Littlewood identity in the setting of
this paper are equivalent to a collection of Whittaker integral identities
conjectured by Bump (1989) and Bump and Friedberg (1990) and proved by Stade
(2001, 2002). Our approach leads to new `combinatorial' proofs and
generalizations of these identities, with some restrictions on the parameters.Comment: v2: significantly extended versio
Models of q-algebra representations: q-integral transforms and "addition theorems''
In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case
Iterative structure of finite loop integrals
In this paper we develop further and refine the method of differential
equations for computing Feynman integrals. In particular, we show that an
additional iterative structure emerges for finite loop integrals. As a concrete
non-trivial example we study planar master integrals of light-by-light
scattering to three loops, and derive analytic results for all values of the
Mandelstam variables and and the mass . We start with a recent
proposal for defining a basis of loop integrals having uniform transcendental
weight properties and use this approach to compute all planar two-loop master
integrals in dimensional regularization. We then show how this approach can be
further simplified when computing finite loop integrals. This allows us to
discuss precisely the subset of integrals that are relevant to the problem. We
find that this leads to a block triangular structure of the differential
equations, where the blocks correspond to integrals of different weight. We
explain how this block triangular form is found in an algorithmic way. Another
advantage of working in four dimensions is that integrals of different loop
orders are interconnected and can be seamlessly discussed within the same
formalism. We use this method to compute all finite master integrals needed up
to three loops. Finally, we remark that all integrals have simple Mandelstam
representations.Comment: 26 pages plus appendices, 5 figure
Matrix models for classical groups and ToeplitzHankel minors with applications to Chern-Simons theory and fermionic models
We study matrix integration over the classical Lie groups
and , using symmetric function theory and the equivalent formulation
in terms of determinants and minors of ToeplitzHankel matrices. We
establish a number of factorizations and expansions for such integrals, also
with insertions of irreducible characters. As a specific example, we compute
both at finite and large the partition functions, Wilson loops and Hopf
links of Chern-Simons theory on with the aforementioned symmetry
groups. The identities found for the general models translate in this context
to relations between observables of the theory. Finally, we use character
expansions to evaluate averages in random matrix ensembles of Chern-Simons
type, describing the spectra of solvable fermionic models with matrix degrees
of freedom.Comment: 32 pages, v2: Several improvements, including a Conclusions and
Outlook section, added. 36 page
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