136 research outputs found
Combinatorial proof for a stability property of plethysm coefficients
Plethysm coefficients are important structural constants in the representation the-
ory of the symmetric groups and general linear groups. Remarkably, some sequences
of plethysm coefficients stabilize (they are ultimately constants). In this paper we
give a new proof of such a stability property, proved by Brion with geometric representation theory techniques. Our new proof is purely combinatorial: we decompose
plethysm coefficients as a alternating sum of terms counting integer points in poly-
topes, and exhibit bijections between these sets of integer points.Ministerio de Ciencia e InnovaciĂłn MTM2010–19336Junta de AndalucĂa FQM–333Junta de AndalucĂa P12–FQM–269
Obstructions to combinatorial formulas for plethysm
Motivated by questions of Mulmuley and Stanley we investigate
quasi-polynomials arising in formulas for plethysm. We demonstrate, on the
examples of and , that these need not be counting
functions of inhomogeneous polytopes of dimension equal to the degree of the
quasi-polynomial. It follows that these functions are not, in general, counting
functions of lattice points in any scaled convex bodies, even when restricted
to single rays. Our results also apply to special rectangular Kronecker
coefficients.Comment: 7 pages; v2: Improved version with further reaching counterexamples;
v3: final version as in Electronic Journal of Combinatoric
Stability properties of Plethysm: new approach with combinatorial proofs (Extended abstract)
International audiencePlethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a newapproach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope.Les coefficients du pléthysme sont des constantes de structure importantes de la théorie des fonctions symétriques, ainsi que de la théorie de la représentation des groupes symétriques et des groupes généraux linéaires. En 1950, Foulkes a observé pour ces coefficients de phénomènes de stabilité: certaines suites de coefficients du pléthysme sont stationnaires. De telles propriétés ont été démontrées par Brion, au moyen de techniques géométriques, et par Thibon et Carré, au moyen d’opérateurs vertex. Dans ce travail, nous présentons une nouvelle approche, purement combinatoire, pour démontrer des propriétés de stabilité de ce type. Nous décomposons les coefficients du pléthysme comme somme alternées de coefficients de pléthysme d’un autre type (liés à la base des fonctions symétriques sommes complètes), qui comptent les points entiers dans des polytopes. Nous démontrons la stabilité des suites de ces coefficients en exhibant des bijections entres les ensembles de points entiers des polytopes correspondants
Brill-Gordan Loci, Transvectants and an Analogue of the Foulkes Conjecture
Combining a selection of tools from modern algebraic geometry, representation
theory, the classical invariant theory of binary forms, together with explicit
calculations with hypergeometric series and Feynman diagrams, we obtain the
following interrelated results. A Castelnuovo-Mumford regularity bound and a
projective normality result for the locus of hypersufaces that are equally
supported on two hyperplanes. The surjectivity of an equivariant map between
two plethystic compositions of symmetric powers; a statement which is
reminiscent of the Foulkes-Howe conjecture. The nonvanishing of even
transvectants of exact powers of generic binary forms. The nonvanishing of a
collection of symmetric functions defined by sums over magic squares and
transportation matrices with nonnegative integer entries. An explicit set of
generators, in degree three, for the ideal of the coincident root locus of
binary forms with only two roots of equal multiplicity.Comment: This is a considerably expanded version of math.AG/040523
Recommended from our members
Mini-Workshop: Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory
The Kronecker, plethysm and Sylow branching coefficients describe the decomposition of representations of symmetric groups obtained by tensor products and induction.
Understanding these decompositions has been hailed as
one of the definitive open problems in algebraic combinatorics and has profound and deep connections with representation theory, symplectic geometry, complexity theory, quantum information theory, and local-global conjectures in representation theory of finite groups.
The overarching theme of the Mini-Workshop has been the use of hidden, richer representation theoretic structures to
prove and disprove conjectures concerning these coefficients.
These structures arise from
the
modular and local-global representation theory of symmetric groups,
graded representation theory of Hecke and Cherednik algebras, and categorical Lie theory
No occurrence obstructions in geometric complexity theory
The permanent versus determinant conjecture is a major problem in complexity
theory that is equivalent to the separation of the complexity classes VP_{ws}
and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a
strengthened version of this conjecture over the complex numbers that amounts
to separating the orbit closures of the determinant and padded permanent
polynomials. In that paper it was also proposed to separate these orbit
closures by exhibiting occurrence obstructions, which are irreducible
representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit
closure, but not in the other. We prove that this approach is impossible.
However, we do not rule out the general approach to the permanent versus
determinant problem via multiplicity obstructions as proposed by Mulmuley and
Sohoni.Comment: Substantial revision. This version contains an overview of the proof
of the main result. Added material on the model of power sums. Theorem 4.14
in the old version, which had a complicated proof, became the easy Theorem
5.4. To appear in the Journal of the AM
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