328 research outputs found
Higher Width Moonshine
\textit{Weak moonshine} for a finite group is the phenomenon where an
infinite dimensional graded -module
has the property that its trace functions, known as McKay-Thompson series, are
modular functions. Recent work by DeHority, Gonzalez, Vafa, and Van Peski
established that weak moonshine holds for every finite group. Since weak
moonshine only relies on character tables, which are not isomorphism class
invariants, non-isomorphic groups can have the same McKay-Thompson series. We
address this problem by extending weak moonshine to arbitrary width
. For each and each irreducible character
, we employ Frobenius' -character extension to define \textit{width McKay-Thompson
series} for ( copies) for each
-tuple in ( copies). These series are
modular functions which then reflect differences between -character values.
Furthermore, we establish orthogonality relations for the Frobenius
-characters, which dictate the compatibility of the extension of weak
moonshine for to width weak moonshine.Comment: Versions 2 and 3 address comments from the referee
Branes, Anti-Branes and Brauer Algebras in Gauge-Gravity duality
We propose gauge theory operators built using a complex Matrix scalar which
are dual to brane-anti-brane systems in , in the zero
coupling limit of the dual Yang-Mills. The branes involved are half-BPS giant
gravitons. The proposed operators dual to giant-anti-giant configurations
satisfy the appropriate orthogonality properties. Projection operators in
Brauer algebras are used to construct the relevant multi-trace Matrix
operators. These are related to the ``coupled representations'' which appear in
2D Yang-Mills theory. We discuss the implications of these results for the
quantum mechanics of a complex matrix model, the counting of non-supersymmetric
operators and the physics of brane-anti-brane systems. The stringy exclusion
principle known from the properties of half-BPS giant gravitons, has a new
incarnation in this context. It involves a qualitative change in the map
between brane-anti-brane states to gauge theory operators. In the case of a
pair of sphere giant and anti-giant this change occurs when the sum of the
magnitudes of their angular momenta reaches .Comment: 52 pages, 10 figure
Set-partition tableaux and representations of diagram algebras
The partition algebra is an associative algebra with a basis of set-partition
diagrams and multiplication given by diagram concatenation. It contains as
subalgebras a large class of diagram algebras including the Brauer, planar
partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook
monoid, and symmetric group algebras. We give a construction of the irreducible
modules of these algebras in two isomorphic ways: first, as the span of
symmetric diagrams on which the algebra acts by conjugation twisted with an
irreducible symmetric group representation and, second, on a basis indexed by
set-partition tableaux such that diagrams in the algebra act combinatorially on
tableaux. The first representation is analogous to the Gelfand model and the
second is a generalization of Young's natural representation of the symmetric
group on standard tableaux. The methods of this paper work uniformly for the
partition algebra and its diagram subalgebras. As an application, we express
the characters of each of these algebras as nonnegative integer combinations of
symmetric group characters whose coefficients count fixed points under
conjugation
Seminormal forms and Gram determinants for cellular algebras
This paper develops an abstract framework for constructing ``seminormal
forms'' for cellular algebras. That is, given a cellular R-algebra A which is
equipped with a family of JM-elements we give a general technique for
constructing orthogonal bases for A, and for all of its irreducible
representations, when the JM-elements separate A. The seminormal forms for A
are defined over the field of fractions of R. Significantly, we show that the
Gram determinant of each irreducible A-module is equal to a product of certain
structure constants coming from the seminormal basis of A. In the non-separated
case we use our seminormal forms to give an explicit basis for a block
decomposition of A.
The appendix, by Marcos Soriano, gives a general construction of a complete
set of orthogonal idempotents for an algera starting from a set of elements
which act on the algebra in an upper triangular fashion. The appendix shows
that constructions with "Jucys-Murphy elements"depend, ultimately, on the
Cayley-Hamilton theorem.Comment: Final version. To appear J. Reine Angew. Math. Appendix by Marcos
Sorian
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