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Permanent and determinant
AbstractThe n×n permanent is not a projection of the m×m determinant if m ⩽ √2n− 6√n
Almost Settling the Hardness of Noncommutative Determinant
In this paper, we study the complexity of computing the determinant of a
matrix over a non-commutative algebra. In particular, we ask the question,
"over which algebras, is the determinant easier to compute than the permanent?"
Towards resolving this question, we show the following hardness and easiness of
noncommutative determinant computation.
* [Hardness] Computing the determinant of an n \times n matrix whose entries
are themselves 2 \times 2 matrices over a field is as hard as computing the
permanent over the field. This extends the recent result of Arvind and
Srinivasan, who proved a similar result which however required the entries to
be of linear dimension.
* [Easiness] Determinant of an n \times n matrix whose entries are themselves
d \times d upper triangular matrices can be computed in poly(n^d) time.
Combining the above with the decomposition theorem of finite dimensional
algebras (in particular exploiting the simple structure of 2 \times 2 matrix
algebras), we can extend the above hardness and easiness statements to more
general algebras as follows. Let A be a finite dimensional algebra over a
finite field with radical R(A).
* [Hardness] If the quotient A/R(A) is non-commutative, then computing the
determinant over the algebra A is as hard as computing the permanent.
* [Easiness] If the quotient A/R(A) is commutative and furthermore, R(A) has
nilpotency index d (i.e., the smallest d such that R(A)d = 0), then there
exists a poly(n^d)-time algorithm that computes determinants over the algebra
A.
In particular, for any constant dimensional algebra A over a finite field,
since the nilpotency index of R(A) is at most a constant, we have the following
dichotomy theorem: if A/R(A) is commutative, then efficient determinant
computation is feasible and otherwise determinant is as hard as permanent.Comment: 20 pages, 3 figure
Symmetries of plane partitions and the permanent-determinant method
In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives
formulas for the number of plane partitions in each of ten symmetry classes.
This paper together with results by Andrews [J. Combin. Theory Ser. A 66
(1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the
project of proving all ten formulas.
We enumerate cyclically symmetric, self-complementary plane partitions. We
first convert plane partitions to tilings of a hexagon in the plane by
rhombuses, or equivalently to matchings in a certain planar graph. We can then
use the permanent-determinant method or a variant, the Hafnian-Pfaffian method,
to obtain the answer as the determinant or Pfaffian of a matrix in each of the
ten cases. We row-reduce the resulting matrix in the case under consideration
to prove the formula. A similar row-reduction process can be carried out in
many of the other cases, and we analyze three other symmetry classes of plane
partitions for comparison
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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