5,071 research outputs found
Uniqueness of Kusuoka Representations
This paper addresses law invariant coherent risk measures and their Kusuoka
representations. By elaborating the existence of a minimal representation we
show that every Kusuoka representation can be reduced to its minimal
representation. Uniqueness -- in a sense specified in the paper -- of the risk
measure's Kusuoka representation is derived from this initial result.
Further, stochastic order relations are employed to identify the minimal
Kusuoka representation. It is shown that measures in the minimal representation
are extremal with respect to the order relations. The tools are finally
employed to provide the minimal representation for important practical
examples. Although the Kusuoka representation is usually given only for
nonatomic probability spaces, this presentation closes the gap to spaces with
atoms
Trees, parking functions, syzygies, and deformations of monomial ideals
For a graph G, we construct two algebras, whose dimensions are both equal to
the number of spanning trees of G. One of these algebras is the quotient of the
polynomial ring modulo certain monomial ideal, while the other is the quotient
of the polynomial ring modulo certain powers of linear forms. We describe the
set of monomials that forms a linear basis in each of these two algebras. The
basis elements correspond to G-parking functions that naturally came up in the
abelian sandpile model. These ideals are instances of the general class of
monotone monomial ideals and their deformations. We show that the Hilbert
series of a monotone monomial ideal is always bounded by the Hilbert series of
its deformation. Then we define an even more general class of monomial ideals
associated with posets and construct free resolutions for these ideals. In some
cases these resolutions coincide with Scarf resolutions. We prove several
formulas for Hilbert series of monotone monomial ideals and investigate when
they are equal to Hilbert series of deformations. In the appendix we discuss
the sandpile model.Comment: 33 pages; v2: appendix on sandpiles added, references added, typos
corrected; v3: references adde
On -Whittaker functions
The -Whittaker functions are eigenfunctions of the modular -deformed
open Toda system introduced by Kharchev, Lebedev, and
Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named
authors obtained a Mellin-Barnes integral representation for these
eigenfunctions. In the present paper, we develop the analytic theory of the
-Whittaker functions from the perspective of quantum cluster algebras. We
obtain a formula for the modular open Toda system's Baxter operator as a
sequence of quantum cluster transformations, and thereby derive a new modular
-analog of Givental's integral formula for the undeformed Whittaker
function. We also show that the -Whittaker functions are eigenvectors of the
Dehn twist operator from quantum higher Teichm\"uller theory, and obtain
-analogs of various integral identities satisfied by the undeformed
Whittaker functions, including the continuous Cauchy-Littlewood identity of
Stade and Corwin-O'Connell-Sepp\"al\"ainen-Zygouras. Using these results, we
prove the unitarity of the -Whittaker transform, thereby completing the
analytic part of the proof of the conjecture of Frenkel and Ip on tensor
products of positive representations of , as well as the
main step in the modular functor conjecture of Fock and Goncharov. We conclude
by explaining how the theory of -Whittaker functions can be used to derive
certain hyperbolic hypergeometric integral evaluations found by Rains.Comment: 36 pages, minor changes, references adde
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