4,013 research outputs found
The number of terms in the permanent and the determinant of a generic circulant matrix
Let A=(a_(ij)) be the generic n by n circulant matrix given by
a_(ij)=x_(i+j), with subscripts on x interpreted mod n. Define d(n) (resp.
p(n)) to be the number of terms in the determinant (resp. permanent) of A. The
function p(n) is well-known and has several combinatorial interpretations. The
function d(n), on the other hand, has not been studied previously. We show that
when n is a prime power, d(n)=p(n). The proof uses symmetric functions.Comment: 6 pages; 1 figur
An analogue of distributivity for ungraded lattices
In this paper, we define a property, trimness, for lattices. Trimness is a
not-necessarily-graded generalization of distributivity; in particular, if a
lattice is trim and graded, it is distributive. Trimness is preserved under
taking intervals and suitable sublattices. Trim lattices satisfy a weakened
form of modularity. The order complex of a trim lattice is contractible or
homotopic to a sphere; the latter holds exactly if the maximum element of the
lattice is a join of atoms.
Other than distributive lattices, the main examples of trim lattices are the
Tamari lattices and various generalizations of them. We show that the Cambrian
lattices in types A and B defined by Reading are trim, and we conjecture that
all Cambrian lattices are trim.Comment: 19 pages, 4 figures. Version 2 includes small improvements to
exposition, corrections of typos, and a new section showing that if a group G
acts on a trim lattice by lattice automorphisms, then the sublattice of L
consisting of elements fixed by G is tri
Cycle-level intersection theory for toric varieties
This paper addresses the problem of constructing a cycle-level intersection
theory for toric varieties. We show that by making one global choice, we can
determine a cycle representative for the intersection of an equivariant Cartier
divisor with an invariant cycle on a toric variety. For a toric variety defined
by a fan in N, the choice consists of giving an inner product or a complete
flag for M_Q=Hom(N,Q), or more generally giving for each cone sigma in the fan
a linear subspace of M_Q complementary to the subspace of M_Q perpendicular to
sigma, satisfying certain compatibility conditions.
We show that these intersection cycles have properties analogous to the usual
intersections modulo rational equivalence. If X is simplicial (for instance, if
X is non-singular), we obtain a commutative ring structure on the invariant
cycles of X with rational coefficients. This ring structure determines cycles
representing certain characteristic classes of the toric variety. We also
discuss how to define intersection cycles that require no choices, at the
expense of increasing the size of the coefficient field.Comment: 24 pages, 3 figure
Defining an m-cluster category
We show that a certain orbit category considerd by Keller encodes the
combinatorics of the -clusters of Fomin and Reading in a fashion similar to
the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov
encodes the combinatorics of the clusters of Fomin and Zelevinsky. This allows
us to give type-uniform proofs of certain results of Fomin and Reading in the
simply laced cases.Comment: Version 2 is substantially shorter, more focussed, and uses less
machinery. Version 3 has minor revisions following referee's suggestions, and
some sign corrections. To appear in J. Alg. 10 page
Graded left modular lattices are supersolvable
We provide a direct proof that a finite graded lattice with a maximal chain
of left modular elements is supersolvable. This result was first established
via a detour through EL-labellings in [McNamara-Thomas] by combining results of
McNamara and Liu. As part of our proof, we show that the maximum graded
quotient of the free product of a chain and a single-element lattice is finite
and distributive.Comment: 7 pages; 2 figures. Version 2: typos and a small error corrected;
diagrams prettier; exposition improved following referee's suggestions;
version to appear in Algebra Universali
Bounding the degrees of generators of a homogeneous dimension 2 toric ideal
Let I be the toric ideal defined by a 2 x n matrix of integers, A = ((1 1 ...
1)(a_1 a_2 ... a_n)) with a_1<a_2<...<a_n. We give a combinatorial proof that I
is generated by elements of degree at most the sum of the two largest
differences a_i - a_(i-1). The novelty is in the method of proof: the result
has already been shown by L'vovsky using cohomological arguments.Comment: 8 pages. To appear in Collectanea Mathematic
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