198 research outputs found
Indecomposable representations and oscillator realizations of the exceptional Lie algebra G_2
In this paper various representations of the exceptional Lie algebra G_2 are
investigated in a purely algebraic manner, and multi-boson/multi-fermion
realizations are obtained. Matrix elements of the master representation, which
is defined on the space of the universal enveloping algebra of G_2, are
explicitly determined. From this master representation, different
indecomposable representations defined on invariant subspaces or quotient
spaces with respect to these invariant subspaces are discussed. Especially, the
elementary representations of G_2 are investigated in detail, and the
corresponding six-boson realization is given. After obtaining explicit forms of
all twelve extremal vectors of the elementary representation with the highest
weight {\Lambda}, all representations with their respective highest weights
related to {\Lambda} are systematically discussed. For one of these
representations the corresponding five-boson realization is constructed.
Moreover, a new three-fermion realization from the fundamental representation
(0,1) of G_2 is constructed also.Comment: 29 pages, 4 figure
A uniform approach to soliton cellular automata using rigged configurations
For soliton cellular automata, we give a uniform description and proofs of
the solitons, the scattering rule of two solitons, and the phase shift using
rigged configurations in a number of special cases. In particular, we prove
these properties for the soliton cellular automata using when is
adjacent to in the Dynkin diagram or there is a Dynkin diagram automorphism
sending to .Comment: 37 pages, 3 figures, 4 table
Some Generalizations of Classical Integer Sequences Arising in Combinatorial Representation Theory
There exists a natural correspondence between the bases for a given finite-dimensional representation of a complex semisimple Lie algebra and a certain collection of finite edge-colored ranked posets, laid out by Donnelly, et al. in, for instance, [Don03]. In this correspondence, the Serre relations on the Chevalley generators of the given Lie algebra are realized as conditions on coeļ¬icients assigned to poset edges. These conditions are the so-called diamond, crossing, and structure relations (hereinafter DCS relations.) New representation constructions of Lie algebras may thus be obtained by utilizing edge-colored ranked posets. Of particular combinatorial interest are those representations whose corresponding posets are distributive lattices. We study two families of such lattices, which we dub the generalized Fibonaccian lattices LFā±įµpn`1, kq and generalized Catalanian lattices LCįµįµpn, kq. These respectively generalize known families of lattices which are DCS-correspondent to some special families of representations of the classical Lie algebras An`ā and Cn. We state and prove explicit formulae for the vertex cardinalities of these lattices; show existence and uniqueness of DCS-satisfactory edge coeļ¬icients for certain values of n and k; and report on the eļ¬icacy of various computational and algorithmic approaches to this problem. A Python library for computationally modeling and āsolvingā these lattices appears as an appendix
Distributive lattice models of the type C one-rowed Weyl group symmetric functions
We present two families of diamond-colored distributive lattices ā one known and one new ā that we can show are models of the type C one-rowed Weyl symmetric functions. These lattices are constructed using certain sequences of positive integers that are visualized as ļ¬lling the boxes of one-rowed partition diagrams. We show how natural orderings of these one-rowed tableaux produce our distributive lattices as sublattices of a more general object, and how a natural coloring of the edges of the associated order diagrams yields a certain diamond-coloring property. We show that each edge-colored lattice possesses a certain structure that is associated with the type C Weyl groups. Moreover, we produce a bijection that shows how any two aļ¬liated lattices, one from each family, are models for the same type C one-rowed Weyl symmetric function. While our type C one-rowed lattices have multiple algebraic contexts, this thesis largely focusses on their combinatorial aspects
From entanglement renormalisation to the disentanglement of quantum double models
We describe how the entanglement renormalisation approach to topological
lattice systems leads to a general procedure for treating the whole spectrum of
these models, in which the Hamiltonian is gradually simplified along a parallel
simplification of the connectivity of the lattice. We consider the case of
Kitaev's quantum double models, both Abelian and non-Abelian, and we obtain a
rederivation of the known map of the toric code to two Ising chains; we pay
particular attention to the non-Abelian models and discuss their space of
states on the torus. Ultimately, the construction is universal for such models
and its essential feature, the lattice simplification, may point towards a
renormalisation of the metric in continuum theories.Comment: 46 pages, 25 eps figure
- ā¦