CAPITAL REGION ALGEBRA/NUMBER THEORY SEMINAR 


Speaker:       Professor Seth Chaiken, Univ. at Albany  

Time:          Wednesday, October 6, at 4:30pm       

Place:         Earth Sciences Bldg. 152A,  Univ. at Albany 

Title:         Extensor Valued Tutte Function on Ported Regular
               Matroids and its Universal Polynomial


Abstract:

Within the exterior algebra of a linear space with given
finite basis (called the ground set S), the indecomposibles
(called extensors) represent linear subspaces, equivalent
systems of equations and their solutions, and hence linearly
representable matroids, both ordinary and oriented.

We formulate the (oriented) matroid operations
deletion  M --> M\e  and contraction   M --> M/e  
of  e in S, and also the (oriented) matroid dual on such 
extensors.

The classical Tutte invariants of matroids take on
commutative values and satisfy Tutte equations:
f(M) = f(M\e) + f(M/e)  and  f(M_1\oplus M_2) = f(M_1)f(M_2).

They derive (for graphic matroids) the spanning tree count, the
chromatic polynomial and the reliability.

After we distinguish a subset  P of elements in E (called ports)
so  P  indicates coordinates onto which a solution space will 
be projected, we define the extensor valued function  M_E(N)  
of extensors  N.

The main result is  M_E(N)  satisfies a sign corrected
anti-commutative variant of the Tutte equations
restricted to an element  e  not in  P, which requires an 
arbitrary orientation of  S  to be specified.

Suppose  N  is specialized to represent a graphic oriented matroid
(regular matroid, more generally). Then:

(1) Our extensor value  M_E(N)  depends only on the oriented matroid
structure independent of the particular representation  N.

(2) M_E(N)  generalizes the determinant of the combinatorial Laplacian
matrix of a graph.  As such it generalizes spanning tree count and expresses
the solutions to Kirchhoffs' and Ohm's laws for electrical currents and
potentials in a graph based network, projected onto the port coordinates.
Hence we explain ``Maxwell's rule'' which expresses equivalent resistance
as a ratio of spanning tree counts.

Like the classical Tutte invariants,  M_E(N)  is an evaluation
(by substituting extensors for variables) of a polynomial that is
universal for all solutions to the Tutte equations
restricted to  e  not in  P  on oriented matroids.  This polynomial
generalizes several known enumerative combinatorial interpretations
of the classical Tutte polynomial based on aspects of matroid
structure to also include some orientation structure.

All the results extend when two parameters we denoted by
g_e, r_e  are given for  e  not in  P  and they are the coefficients
in the additive Tutte equation.  The electrical interpretation
is  r_e:g_e  equals the resistance in network edge  e.



Refreshments at 4:15 pm in ES 152.