147 research outputs found
A max-flow algorithm for positivity of Littlewood-Richardson coefficients
Littlewood-Richardson coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group . They have a wide variety of interpretations in combinatorics, representation theory and geometry. Mulmuley and Sohoni pointed out that it is possible to decide the positivity of Littlewood-Richardson coefficients in polynomial time. This follows by combining the saturation property of Littlewood-Richardson coefficients (shown by Knutson and Tao 1999) with the well-known fact that linear optimization is solvable in polynomial time. We design an explicit polynomial time algorithm for deciding the positivity of Littlewood-Richardson coefficients. This algorithm is highly adapted to the problem and it is based on ideas from the theory of optimizing flows in networks
On vanishing of Kronecker coefficients
We show that the problem of deciding positivity of Kronecker coefficients is
NP-hard. Previously, this problem was conjectured to be in P, just as for the
Littlewood-Richardson coefficients. Our result establishes in a formal way that
Kronecker coefficients are more difficult than Littlewood-Richardson
coefficients, unless P=NP.
We also show that there exists a #P-formula for a particular subclass of
Kronecker coefficients whose positivity is NP-hard to decide. This is an
evidence that, despite the hardness of the positivity problem, there may well
exist a positive combinatorial formula for the Kronecker coefficients. Finding
such a formula is a major open problem in representation theory and algebraic
combinatorics.
Finally, we consider the existence of the partition triples such that the Kronecker coefficient but the
Kronecker coefficient for some integer
. Such "holes" are of great interest as they witness the failure of the
saturation property for the Kronecker coefficients, which is still poorly
understood. Using insight from computational complexity theory, we turn our
hardness proof into a positive result: We show that not only do there exist
many such triples, but they can also be found efficiently. Specifically, we
show that, for any , there exists such that, for all
, there exist partition triples in the
Kronecker cone such that: (a) the Kronecker coefficient
is zero, (b) the height of is , (c) the height of is , and (d) . The proof of the last result
illustrates the effectiveness of the explicit proof strategy of GCT.Comment: 43 pages, 1 figur
All Kronecker coefficients are reduced Kronecker coefficients
We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of a question by Stanley from 2000 and a question by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is NP-hard, and computing them is #P-hard under parsimonious many-one reductions
On the complexity of computing Kronecker coefficients
We study the complexity of computing Kronecker coefficients
. We give explicit bounds in terms of the number of parts
in the partitions, their largest part size and the smallest second
part of the three partitions. When , i.e. one of the partitions
is hook-like, the bounds are linear in , but depend exponentially on
. Moreover, similar bounds hold even when . By a separate
argument, we show that the positivity of Kronecker coefficients can be decided
in time for a bounded number of parts and without
restriction on . Related problems of computing Kronecker coefficients when
one partition is a hook, and computing characters of are also considered.Comment: v3: incorporated referee's comments; accepted to Computational
Complexit
Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals
Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in
\cite{Chi08}, we use quiver theory to represent the generalized
Littlewood-Richardson coefficients for the branching rule for the diagonal
embedding of \gl(n) as the dimension of a weight space of semi-invariants.
Using this, we prove their saturation and investigate when they are nonzero. We
also show that for certain partitions the associated stretched polynomials
satisfy the same conjectures as single Littlewood-Richardson coefficients. We
then provide a polytopal description of this multiplicity and show that its
positivity may be computed in strongly polynomial time. Finally, we remark that
similar results hold for certain other generalized Littlewood-Richardson
coefficients.Comment: 28 pages, comments welcom
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
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