100 research outputs found
Twisting q-holonomic sequences by complex roots of unity
A sequence is -holonomic if it satisfies a nontrivial linear
recurrence with coefficients polynomials in and . Our main theorems
state that -holonomicity is preserved under twisting, i.e., replacing by
where is a complex root of unity, and under the
substitution where is a rational number. Our proofs
are constructive, work in the multivariate setting of -finite
sequences and are implemented in the Mathematica package HolonomicFunctions.
Our results are illustrated by twisting natural -holonomic sequences which
appear in quantum topology, namely the colored Jones polynomial of pretzel
knots and twist knots. The recurrence of the twisted colored Jones polynomial
can be used to compute the asymptotics of the Kashaev invariant of a knot at an
arbitrary complex root of unity.Comment: 8 pages, 2 figures, 1 table, final version for the ISSAC proceedings;
Proceedings of ISSAC 201
Microlocal KZ functors and rational Cherednik algebras
Following the work of Kashiwara-Rouquier and Gan-Ginzburg, we define a family
of exact functors from category for the rational Cherednik algebra
in type to representations of certain "coloured braid groups" and calculate
the dimensions of the representations thus obtained from standard modules. To
show that our constructions also make sense in a more general context, we also
briefly study the case of the rational Cherednik algebra corresponding to
complex reflection group .Comment: Revised to improve exposition, giving more details on the
construction of the microlocal local systems and providing background
information on twisted D-modules in an appendi
Dualities and non-Abelian mechanics
Dualities are mathematical mappings that reveal unexpected links between
apparently unrelated systems or quantities in virtually every branch of
physics. Systems that are mapped onto themselves by a duality transformation
are called self-dual and they often exhibit remarkable properties, as
exemplified by an Ising magnet at the critical point. In this Letter, we unveil
the role of dualities in mechanics by considering a family of so-called twisted
Kagome lattices. These are reconfigurable structures that can change shape
thanks to a collapse mechanism easily illustrated using LEGO. Surprisingly,
pairs of distinct configurations along the mechanism exhibit the same spectrum
of vibrational modes. We show that this puzzling property arises from the
existence of a duality transformation between pairs of configurations on either
side of a mechanical critical point. This critical point corresponds to a
self-dual structure whose vibrational spectrum is two-fold degenerate over the
entire Brillouin zone. The two-fold degeneracy originates from a general
version of Kramers theorem that applies to classical waves in addition to
quantum systems with fermionic time-reversal invariance. We show that the
vibrational modes of the self-dual mechanical systems exhibit non-Abelian
geometric phases that affect the semi-classical propagation of wave packets.
Our results apply to linear systems beyond mechanics and illustrate how
dualities can be harnessed to design metamaterials with anomalous symmetries
and non-commuting responses.Comment: See http://home.uchicago.edu/~vitelli/videos.html for Supplementary
Movi
Vertex operators, solvable lattice models and metaplectic Whittaker functions
We show that spherical Whittaker functions on an -fold cover of the
general linear group arise naturally from the quantum Fock space representation
of introduced by Kashiwara, Miwa and Stern
(KMS). We arrive at this connection by reconsidering solvable lattice models
known as `metaplectic ice' whose partition functions are metaplectic Whittaker
functions. First, we show that a certain Hecke action on metaplectic Whittaker
coinvariants agrees (up to twisting) with a Hecke action of Ginzburg,
Reshetikhin, and Vasserot. This allows us to expand the framework of KMS by
Drinfeld twisting to introduce Gauss sums into the quantum wedge, which are
necessary for connections to metaplectic forms. Our main theorem interprets the
row transfer matrices of this ice model as `half' vertex operators on quantum
Fock space that intertwine with the action of
.
In the process, we introduce new symmetric functions termed
\textit{metaplectic symmetric functions} and explain how they relate to
Whittaker functions on an -fold metaplectic cover of GL. These resemble
\textit{LLT polynomials} introduced by Lascoux, Leclerc and Thibon; in fact the
metaplectic symmetric functions are (up to twisting) specializations of
\textit{supersymmetric LLT polynomials} defined by Lam. Indeed Lam constructed
families of symmetric functions from Heisenberg algebra actions on the Fock
space commuting with the -action. We explain
that half vertex operators agree with Lam's construction and this
interpretation allows for many new identities for metaplectic symmetric and
Whittaker functions, including Cauchy identities. While both metaplectic
symmetric functions and LLT polynomials can be related to vertex operators on
the -Fock space, only metaplectic symmetric functions are connected to
solvable lattice models.Comment: v3 changes: minor edit
Fast Computation of the -th Term of a -Holonomic Sequence and Applications
33 pages. Long version of the conference paper Computing the -th term of a -holonomic sequence. Proceedings ISSAC'20, pp. 46–53, ACM Press, 2020 (https://hal.inria.fr/hal-02882885)International audienceIn 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial in arithmetic complexity quasi-linear in . In 1988, the Chudnovsky brothers generalized Strassen’s algorithm to the computation of the -th term of any holonomic sequence in essentially the same arithmetic complexity. We design -analogues of these algorithms. We first extend Strassen’s algorithm to the computation of the -factorial of , then Chudnovskys' algorithm to the computation of the -th term of any -holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in ; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear -differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost
Knots, perturbative series and quantum modularity
We introduce an invariant of a hyperbolic knot which is a map \alpha\mapsto
\mathbf{Phi}_\a(h) from to matrices with entries in
and with rows and columns indexed by the boundary
parabolic representations of the fundamental group of the
knot. These matrix invariants have a rich structure: (a) their
entry, where is the trivial and the
geometric representation, is the power series expansion of the Kashaev
invariant of the knot around the root of unity as an
element of the Habiro ring, and the remaining entries belong to generalized
Habiro rings of number fields; (b) the first column is given by the
perturbative power series of Dimofte--Garoufalidis; (c)~the columns of
are fundamental solutions of a linear -difference equation;
(d)~the matrix defines an -cocycle in
matrix-valued functions on that conjecturally extends to a smooth
function on and even to holomorphic functions on suitable complex
cut planes, lifting the factorially divergent series to
actual functions. The two invariants and are
related by a refined quantum modularity conjecture which we illustrate in
detail for the three simplest hyperbolic knots, the , and
pretzel knots. This paper has two sequels, one giving a different realization
of our invariant as a matrix of convergent -series with integer coefficients
and the other studying its Habiro-like arithmetic properties in more depth.Comment: 97 pages, 8 figure
The finiteness conjecture for skein modules
We give a new, algebraically computable formula for skein modules of closed
3-manifolds via Heegaard splittings. As an application, we prove that skein
modules of closed 3-manifolds are finite-dimensional, resolving in the
affirmative a conjecture of Witten.Comment: 38 pages; v3: minor corrections in Section
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