Annals of Combinatorics 1 (1997) 105-106
Combinatorics and Nonparametric Mathematics
Joseph P.S. Kung
Department of Mathematics, University of North Texas, Denton, Texas
Received June 3, 1997
Nonparametric statistics, toric varieties, matroids -- a more disparate trio
of mathematical subjects can hardly be imagined, and yet, they share a basic
idea. The idea is to replace a numerical or continuous quantity in an
existing "classical" subject by a discrete combinatorial quality.
In nonparametric statistics, one replaces the values of the sample points,
which are usually real numbers, by, say, their ranks or signs.
In the theory of toric varieties, the replacement is done
through the action of the group of diagonal matrices. Under this action,
two nonzero coefficients in a polynomial are in the same orbit and,
hence, it only matters whether a coefficient is zero or nonzero.
In matroid theory, one replaces the value of a determinant by
the bit of information whether it is zero or nonzero.
Retaining more information, such as the trinary piece, or
trit of information
whether the determinant is zero,
positive or negative, one obtains oriented matroid theory.
Pushing this further, by retaining a 2, 4, or 8-tuple
of bits or trits, one obtains complex, quarternionic, and octonionic matroids
or oriented matroids.
The last two types of matroids have not been studied and one expects that
the theory will be complicated. A satisfactory
quarternionic matroid would require understanding identities between
over skew fields. Similarly, a theory of octonionic matroids
would uncover the precise role played by associativity in
linear algebra or, equivalently, Desargues' theorem in geometry.
Several papers in this journal show nonparametrization in action.
Four of these papers are about symplectic analogues of matroids.
The paper by Serganova, Vince, and Zelevinsky 
the two papers of Borovik, Gelfand, Vince, and White [1,2] (one of which appeared in
an earlier issue)
study symplectic matroids in terms of a certain polytope similar to the
independent-set polytope for matroids. The more general polytope,
defined using Coxeter groups, was
defined by Gelfand and Serganova  in 1987. The independent-set polytope,
has wide applications in combinatorial optimization since its
definition by Edmonds in 1970 .
Another paper , by the author, gives a direct (and somewhat different)
a symplectic matroid in terms of an exchange-augmentation axiom derived from
a straightening identity for Pfaffians. This definition harks back to
Whitney's definition of a matroid in 1935.
The study of symplectic matroids may be viewed
as part of an Erlanger Programm to study analogues of
matroids for the classical
linear groups first
suggested by Gian-Carlo Rota in his Bowdoin lectures in 1971 .
Symplectic matroids also provide a natural context to study the
synthetic or intrinsic geometry of symplectic spaces.
Another way combinatorics can assume center stage in a subject
is to replace a continuous object by simpler combinatorial structures.
The classical example of this is the replacement of a topological
space by a simplicial complex in algebraic topology. The
definition of combinatorial differential manifolds,
by Gelfand and MacPherson, is a more recent example. Many properties
arrangements of hyperplanes reside in the matroid determined by
its lattice of intersection.
The paper by Michael Falk ,
is about connections between
an arrangement and its matroid. More specifically, it studies the
question whether the Orlik-Solomon algebra of
a matroid determines that matroid and gives an account of what is known
in the rank-3 case.
The paper of Terhalle , studies the role played by a valuated matroid in
These papers give an indication why
many have considered
combinatorics to be one of the central subjects of mathematics.
A.V. Borovik, I.M. Gelfand, A. Vince, and N.L. White,
The lattice of flats and its
underlying flag matroid polytope,
Ann. Combin. 1 (1997) 17–26.
A.V. Borovik, I.M. Gelfand, and N.L. White,
Coxeter matroid polytopes, Ann. Combin. 1 (1997) 123–134.
3. J. Edmonds, Submodular functions, matroids, and
certain polyhedra, In: Combinatorial Structures and Their Applications,
R. Guy et al., Eds., Gordon and Breach, New York, 1970, pp. 69–87.
4. M. Falk, Arrangements and cohomology, Ann. Combin.
1 (1997) 135–158.
5. I.M. Gelfand and
V.V. Serganova, On a general definition of a matroid and a
greedoid, Soviet Math. Dokl. 35 (1987) 6–10.
J.P.S. Kung, Pfaffian structures and critical problems in finite
symplectic spaces, Ann. Combin. 1 (1997) 159–172.
7. G.-C. Rota,
Combinatorial Theory and Invariant Theory,
Notes taken by L. Guibas from the National Science Foundation
Seminar in Combinatorial Theory, Bowdoin College, Maine,
1971, unpublished typescript.
V.V. Serganova, A. Vince, and A. Zelevinsky,
A geometric characterization of Coxeter matroids, Ann. Combin. 1
W.F. Terhalle, Coordinatizing R-trees in terms of Universal c-trees,
Ann. Combin. 1 (1997) 183–196.
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