66 research outputs found
Rectangular Kronecker coefficients and plethysms in geometric complexity theory
We prove that in the geometric complexity theory program the vanishing of
rectangular Kronecker coefficients cannot be used to prove superpolynomial
determinantal complexity lower bounds for the permanent polynomial.
Moreover, we prove the positivity of rectangular Kronecker coefficients for a
large class of partitions where the side lengths of the rectangle are at least
quadratic in the length of the partition. We also compare rectangular Kronecker
coefficients with their corresponding plethysm coefficients, which leads to a
new lower bound for rectangular Kronecker coefficients. Moreover, we prove that
the saturation of the rectangular Kronecker semigroup is trivial, we show that
the rectangular Kronecker positivity stretching factor is 2 for a long first
row, and we completely classify the positivity of rectangular limit Kronecker
coefficients that were introduced by Manivel in 2011.Comment: 20 page
Even Partitions in Plethysms
We prove that for all natural numbers k,n,d with k <= d and every partition
lambda of size kn with at most k parts there exists an irreducible GL(d,
C)-representation of highest weight 2*lambda in the plethysm Sym^k(Sym^(2n)
(C^d)). This gives an affirmative answer to a conjecture by Weintraub (J.
Algebra, 129 (1):103-114, 1990). Our investigation is motivated by questions of
geometric complexity theory and uses ideas from quantum information theory.Comment: 9 page
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
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
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
The stability of the Kronecker products of Schur functions
In the late 1930's Murnaghan discovered the existence of a stabilization
phenomenon for the Kronecker product of Schur functions. For n sufficiently
large, the values of the Kronecker coefficients appearing in the product of two
Schur functions of degree n do not depend on the first part of the indexing
partitions, but only on the values of their remaining parts. We compute the
exact value of n for which all the coefficients of a Kronecker product of Schur
functions stabilize. We also compute two new bounds for the stabilization of a
sequence of coefficients and show that they improve existing bounds of M. Brion
and E. Vallejo.Comment: 16 page
Rectangular symmetries for coefficients of symmetric functions
We show that some of the main structural constants for symmetric functions
(Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.Ministerio de Ciencia e InnovaciónJunta de AndalucíaNational Science Foundation Gran
Breaking down the reduced Kronecker coefficients
We resolve three interrelated problems on \emph{reduced Kronecker
coefficients} . First, we disprove the
\emph{saturation property} which states that
implies
for all . Second, we esimate the
maximal , over all
. Finally, we show that computing
is strongly -hard, i.e. -hard when
the input is in unary.Comment: 5 page
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