8,290 research outputs found
Weighted Khovanov-Lauda-Rouquier algebras
In this paper, we define a generalization of Khovanov-Lauda-Rouquier algebras
which we call weighted Khovanov-Lauda-Rouquier algebras. We show that these
algebras carry many of the same structures as the original
Khovanov-Lauda-Rouquier algebras, including induction and restriction functors
which induce a twisted biaglebra structure on their Grothendieck groups.
We also define natural quotients of these algebras, which in an important
special case carry a categorical action of an associated Lie algebra. Special
cases of these include the algebras categorifying tensor products and Fock
spaces defined by the author and Stroppel in past work.
For symmetric Cartan matrices, weighted KLR algebras also have a natural
gometric interpretation as convolution algebras, generalizing that for the
original KLR algebras by Varagnolo and Vasserot; this result has positivity
consequences important in the theory of crystal bases. In this case, we can
also relate the Grothendieck group and its bialgebra structure to the Hall
algebra of the associated quiver.Comment: 37 pages; v4: edits in response to referee report. Biggest change is
characteristic 0 assumption in Section
Graphs and networks theory
This chapter discusses graphs and networks theory
A norm for the cohomology of 2-complexes
We introduce a norm on the real 1-cohomology of finite 2-complexes determined
by the Euler characteristics of graphs on these complexes. We also introduce
twisted Alexander-Fox polynomials of groups and show that they give rise to
norms on the real 1-cohomology of groups. Our main theorem states that for a
finite 2-complex X, the norm on H^1(X; R) determined by graphs on X majorates
the Alexander-Fox norms derived from \pi_1(X).Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-7.abs.htm
The Relativized Second Eigenvalue Conjecture of Alon
We prove a relativization of the Alon Second Eigenvalue Conjecture for all
-regular base graphs, , with : for any , we show that
a random covering map of degree to has a new eigenvalue greater than
in absolute value with probability .
Furthermore, if is a Ramanujan graph, we show that this probability is
proportional to , where
is an integer depending on , which can be computed by a finite algorithm for
any fixed . For any -regular graph, , is
greater than .
Our proof introduces a number of ideas that simplify and strengthen the
methods of Friedman's proof of the original conjecture of Alon. The most
significant new idea is that of a ``certified trace,'' which is not only
greatly simplifies our trace methods, but is the reason we can obtain the
estimate above. This estimate represents an
improvement over Friedman's results of the original Alon conjecture for random
-regular graphs, for certain values of
Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs
We present an asymptotic expansion for quaternionic self-adjoint matrix
integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon
graphs and their non-orientable counterparts. The result exhibits a striking
duality between quaternionic self-adjoint and real symmetric matrix integrals.
The asymptotic expansions of these integrals are given in terms of summations
over topologies of compact surfaces, both orientable and non-orientable, for
all genera and an arbitrary positive number of marked points on them. We show
that the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble
(GSE) have exactly the same graphical expansion term by term (when
appropriately normalized),except that the contributions from non-orientable
surfaces with odd Euler characteristic carry the opposite sign. As an
application, we give a new topological proof of the known duality for
correlations of characteristic polynomials. Indeed, we show that this duality
is equivalent to Poincare duality of graphs drawn on a compact surface. Another
application of our graphical expansion formula is a simple and simultaneous
(re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary
Ensemble) and GSE: The three cases have exactly the same graphical limiting
formula except for an overall constant that represents the type of the
ensemble.Comment: 39 pages, AMS LaTeX, 49 .eps figures, references update
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