1,460 research outputs found
An observation on highest weight crystals
As shown by Stembridge, crystal graphs can be characterized by their local
behavior. In this paper, we observe that a certain local property on highest
weight crystals forces a more global property. In type , this statement says
that if a node has a single parent and single grandparent, then there is a
unique walk from the highest weight node to it. In other classical types, there
is a similar (but necessarily more technical) statement. This walk is obtained
from the associated level 1 perfect crystal, . (It is unique unless
the Dynkin diagram contains that of as a subdiagram.)
This crystal observation was motivated by representation-theoretic behavior
of the affine Hecke algebra of type , which is known to be captured by
highest weight crystals of type by results of Grojnowski. As
discussed below, the proofs in either setting are straightforward, and so the
theorem linking the two phenomena is not needed. However, the result is
presented here for crystals as one can say something in all types (Grojnowski's
theorem is only in type ), and because the statement seems more surprising
in the language of crystals than it does for affine Hecke algebra modules
Categorifying the tensor product of a level 1 highest weight and perfect crystal in type A
We use Khovanov-Lauda-Rouquier algebras to categorify a crystal isomorphism
between a highest weight crystal and the tensor product of a perfect crystal
and another highest weight crystal, all in level 1 type A affine. The nodes of
the perfect crystal correspond to a family of trivial modules and the nodes of
the highest weight crystal correspond to simple modules, which we may also
parameterize by -restricted partitions. In the case is a prime,
one can reinterpret all the results for the symmetric group in characteristic
. The crystal operators correspond to socle of restriction and behave
compatibly with the rule for tensor product of crystal graphs.Comment: 29 pages; to appear in Proc. Sympos. Pure Math. as part of the
Proceedings of the 2012-2014 Southeastern Lie Theory Workshop
Non-Separable, Quasiconcave Utilities are Easy -- in a Perfect Price Discrimination Market Model
Recent results, establishing evidence of intractability for such restrictive
utility functions as additively separable, piecewise-linear and concave, under
both Fisher and Arrow-Debreu market models, have prompted the question of
whether we have failed to capture some essential elements of real markets,
which seem to do a good job of finding prices that maintain parity between
supply and demand.
The main point of this paper is to show that even non-separable, quasiconcave
utility functions can be handled efficiently in a suitably chosen, though
natural, realistic and useful, market model; our model allows for perfect price
discrimination. Our model supports unique equilibrium prices and, for the
restriction to concave utilities, satisfies both welfare theorems
Categorifying the tensor product of the Kirillov-Reshetikhin crystal and a fundamental crystal
We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal
isomorphism between a fundamental crystal and the tensor product of a
Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine
type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of
"trivial" modules. The nodes of the fundamental crystal correspond to simple
modules of the corresponding cyclotomic KLR algebra. The crystal operators
correspond to socle of restriction and behave compatibly with the rule for
tensor product of crystal graphs.Comment: 58 pages, 4 figures, 4 table
An Incentive Compatible, Efficient Market for Air Traffic Flow Management
We present a market-based approach to the Air Traffic Flow Management (ATFM)
problem. The goods in our market are delays and buyers are airline companies;
the latter pay money to the FAA to buy away the desired amount of delay on a
per flight basis. We give a notion of equilibrium for this market and an LP
whose solution gives an equilibrium allocation of flights to landing slots as
well as equilibrium prices for the landing slots. Via a reduction to matching,
we show that this equilibrium can be computed combinatorially in strongly
polynomial time. Moreover, there is a special set of equilibrium prices, which
can be computed easily, that is identical to the VCG solution, and therefore
the market is incentive compatible in dominant strategy.Comment: arXiv admin note: substantial text overlap with arXiv:1109.521
- β¦