17 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
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
Breaking down the reduced Kronecker coefficients
We resolve three interrelated problems on reduced Kronecker coefficients . First, we disprove the 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
Geometric complexity theory and matrix powering
Valiant's famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a process called padding) and then comparing the orbit closures of the determinant and the padded permanent. This padding was recently used heavily to show no-go results for the method of shifted partial derivatives (Efremenko, Landsberg, Schenck, Weyman, 2016) and for geometric complexity theory (Ikenmeyer Panova, FOCS 2016 and B\"urgisser, Ikenmeyer Panova, FOCS 2016). Following a classical homogenization result of Nisan (STOC 1991) we replace the determinant in geometric complexity theory with the trace of a variable matrix power. This gives an equivalent but much cleaner homogeneous formulation of geometric complexity theory in which the padding is removed. This radically changes the representation theoretic questions involved to prove complexity lower bounds. We prove that in this homogeneous formulation there are no orbit occurrence obstructions that prove even superlinear lower bounds on the complexity of the permanent. This is the first no-go result in geometric complexity theory that rules out superlinear lower bounds in some model. Interestingly---in contrast to the determinant---the trace of a variable matrix power is not uniquely determined by its stabilizer
On the Complexity of the Permanent in Various Computational Models
We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation
The classification of multiplicity-free products of Schur functions
We classify and construct all multiplicity-free plethystic products of Schur functions. We also compute many new (infinite) families of plethysm coefficients, with particular emphasis on those near maximal in the dominance ordering and those of small Durfee size
The classification of multiplicity-free plethysms of Schur functions
We classify and construct all multiplicity-free plethystic products of Schur
functions. We also compute many new (infinite) families of plethysm
coefficients, with particular emphasis on those near maximal in the dominance
ordering and those of small Durfee size.Comment: Extra exposition and examples have been added throughout the first 4
section