21 research outputs found
Computing Multiplicities of Lie Group Representations
For fixed compact connected Lie groups H \subseteq G, we provide a polynomial
time algorithm to compute the multiplicity of a given irreducible
representation of H in the restriction of an irreducible representation of G.
Our algorithm is based on a finite difference formula which makes the
multiplicities amenable to Barvinok's algorithm for counting integral points in
polytopes.
The Kronecker coefficients of the symmetric group, which can be seen to be a
special case of such multiplicities, play an important role in the geometric
complexity theory approach to the P vs. NP problem. Whereas their computation
is known to be #P-hard for Young diagrams with an arbitrary number of rows, our
algorithm computes them in polynomial time if the number of rows is bounded. We
complement our work by showing that information on the asymptotic growth rates
of multiplicities in the coordinate rings of orbit closures does not directly
lead to new complexity-theoretic obstructions beyond what can be obtained from
the moment polytopes of the orbit closures. Non-asymptotic information on the
multiplicities, such as provided by our algorithm, may therefore be essential
in order to find obstructions in geometric complexity theory.Comment: 10 page
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
The Saxl Conjecture and the Dominance Order
In 2012 Jan Saxl conjectured that all irreducible representations of the
symmetric group occur in the decomposition of the tensor square of the
irreducible representation corresponding to the staircase partition. We make
progress on this conjecture by proving the occurrence of all those irreducibles
which correspond to partitions that are comparable to the staircase partition
in the dominance order. Moreover, we use our result to show the occurrence of
all irreducibles corresponding to hook partitions. This generalizes results by
Pak, Panova, and Vallejo from 2013.Comment: 11 page
The border support rank of two-by-two matrix multiplication is seven
We show that the border support rank of the tensor corresponding to
two-by-two matrix multiplication is seven over the complex numbers. We do this
by constructing two polynomials that vanish on all complex tensors with format
four-by-four-by-four and border rank at most six, but that do not vanish
simultaneously on any tensor with the same support as the two-by-two matrix
multiplication tensor. This extends the work of Hauenstein, Ikenmeyer, and
Landsberg. We also give two proofs that the support rank of the two-by-two
matrix multiplication tensor is seven over any field: one proof using a result
of De Groote saying that the decomposition of this tensor is unique up to
sandwiching, and another proof via the substitution method. These results
answer a question asked by Cohn and Umans. Studying the border support rank of
the matrix multiplication tensor is relevant for the design of matrix
multiplication algorithms, because upper bounds on the border support rank of
the matrix multiplication tensor lead to upper bounds on the computational
complexity of matrix multiplication, via a construction of Cohn and Umans.
Moreover, support rank has applications in quantum communication complexity
Complexity of short Presburger arithmetic
We study complexity of short sentences in Presburger arithmetic (Short-PA).
Here by "short" we mean sentences with a bounded number of variables,
quantifiers, inequalities and Boolean operations; the input consists only of
the integers involved in the inequalities. We prove that assuming Kannan's
partition can be found in polynomial time, the satisfiability of Short-PA
sentences can be decided in polynomial time. Furthermore, under the same
assumption, we show that the numbers of satisfying assignments of short
Presburger sentences can also be computed in polynomial time
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
Plethysm and lattice point counting
We apply lattice point counting methods to compute the multiplicities in the
plethysm of . Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition
of 3,4, or 5 we obtain an explicit formula in and for the
multiplicity of in .Comment: 25 pages including appendix, 1 figure, computational results and code
available at http://thomas-kahle.de/plethysm.html, v2: various improvements,
v3: final version appeared in JFoC
Bounds on Kronecker and -binomial coefficients
We present a lower bound on the Kronecker coefficients for tensor squares of
the symmetric group via the characters of~, which we apply to obtain
various explicit estimates. Notably, we extend Sylvester's unimodality of
-binomial coefficients as polynomials in~ to derive
sharp bounds on the differences of their consecutive coefficients. We then
derive effective asymptotic lower bounds for a wider class of Kronecker
coefficients.Comment: version May 2016: improved the effective constants. To appear in
JCTA. This paper is an extension of parts of the earlier paper "Bounds on the
Kronecker coefficients" arXiv:1406.2988, which also contains stability
result
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