1,933 research outputs found
Sextonians and the magic square
Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an additional row and column. The extra row and column in the magic square correspond to the sextonions. This is a six-dimensional subalgebra of the split octonions which contains the split quaternions
Invariant tensors and cellular categories
Let U be the quantised enveloping algebra associated to a Cartan matrix of
finite type. Let W be the tensor product of a finite list of highest weight
representations of U. Then the centraliser algebra of W has a basis called the
dual canonical basis which gives an integral form. We show that this integral
form is cellular by using results due to Lusztig.Comment: 6 pages; to appear in Journal of Algebr
Invariant tensors and the cyclic sieving phenomenon
We construct a large class of examples of the cyclic sieving phenomenon by
expoiting the representation theory of semi-simple Lie algebras. Let be a
finite dimensional representation of a semi-simple Lie algebra and let be
the associated Kashiwara crystal. For , the triple which
exhibits the cyclic sieving phenomenon is constructed as follows: the set
is the set of isolated vertices in the crystal ; the map is a generalisation of promotion acting on standard tableaux of
rectangular shape and the polynomial is the fake degree of the Frobenius
character of a representation of related to the natural action
of on the subspace of invariant tensors in .
Taking to be the defining representation of gives the
cyclic sieving phenomenon for rectangular tableaux
Promotion on oscillating and alternating tableaux and rotation of matchings and permutations
Using Henriques' and Kamnitzer's cactus groups, Sch\"utzenberger's promotion
and evacuation operators on standard Young tableaux can be generalised in a
very natural way to operators acting on highest weight words in tensor products
of crystals.
For the crystals corresponding to the vector representations of the
symplectic groups, we show that Sundaram's map to perfect matchings intertwines
promotion and rotation of the associated chord diagrams, and evacuation and
reversal. We also exhibit a map with similar features for the crystals
corresponding to the adjoint representations of the general linear groups.
We prove these results by applying van Leeuwen's generalisation of Fomin's
local rules for jeu de taquin, connected to the action of the cactus groups by
Lenart, and variants of Fomin's growth diagrams for the Robinson-Schensted
correspondence
Feast, famine and fraud: Considerations in the delivery of banking and financial services to remote Indigenous communities
This discussion paper identifies the current barriers to improving delivery of financial services to indigenous people in Central Australia. It describes existing responses to addressing these barriers, and recommends short- and longer-term options for building on existing initiatives, drawing on demonstrated experience elsewhere in the Northern Territory, Australia and overseas. Many of these issues were identified and discussed at a workshop convened jointly by the Aboriginal and Torres Strait Islander Commission (ATSIC) and Centrelink in Alice Springs on 19 April 1999 with participation drawn from indigenous organisations, banks, indigenous community stores, local government, credit unions and government service deliverers
Series of nilpotent orbits
We organize the nilpotent orbits in the exceptional complex Lie algebras into
series using the triality model and show that within each series the dimension
of the orbit is a linear function of the natural parameter a=1,2,4,8,
respectively for f_4,e_6,e_7,e_8. We also obtain explicit representatives in a
uniform manner. We observe similar regularities for the centralizers of
nilpotent elements in a series and graded components in the associated grading
of the ambient Lie algebra. More strikingly, for a greater than one, the
degrees of the unipotent characters of the corresponding Chevalley groups,
associated to these series through the Springer correspondance are given by
polynomials which have uniform expressions in terms of a.Comment: 20 pages, revised version with more formulas for unipotent character
The equality of 3-manifold invariants
The invariants of 3-manifolds defined by Kuperberg for involutory Hopf
algebras and those defined by the authors for spherical Hopf algebras are the
same for Hopf algebras on which they are both defined.Comment: 8 pages, definition of state sum invariant improved for clarity, plus
minor typos corrected. With 3 postscript figures. further change: BoxedEPSF
macro now include
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