214 research outputs found
A sagbi basis for the quantum Grassmannian
The maximal minors of a p by (m + p) matrix of univariate polynomials of
degree n with indeterminate coefficients are themselves polynomials of degree
np. The subalgebra generated by their coefficients is the coordinate ring of
the quantum Grassmannian, a singular compactification of the space of rational
curves of degree np in the Grassmannian of p-planes in (m + p)-space. These
subalgebra generators are shown to form a sagbi basis. The resulting flat
deformation from the quantum Grassmannian to a toric variety gives a new
`Gr\"obner basis style' proof of the Ravi-Rosenthal-Wang formulas in quantum
Schubert calculus. The coordinate ring of the quantum Grassmannian is an
algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and
Koszul, and the ideal of quantum Pl\"ucker relations has a quadratic Gr\"obner
basis. This holds more generally for skew quantum Schubert varieties. These
results are well-known for the classical Schubert varieties (n=0). We also show
that the row-consecutive p by p-minors of a generic matrix form a sagbi basis
and we give a quadratic Gr\"obner basis for their algebraic relations.Comment: 18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of
Gian-Carlo Rot
Whitney algebras and Grassmann's regressive products
Geometric products on tensor powers of an exterior
algebra and on Whitney algebras \cite{crasch} provide a rigorous version of
Grassmann's {\it regressive products} of 1844 \cite{gra1}. We study geometric
products and their relations with other classical operators on exterior
algebras, such as the Hodge operators and the {\it join} and {\it meet}
products in Cayley-Grassmann algebras \cite{BBR, Stew}. We establish encodings
of tensor powers and of Whitney algebras in
terms of letterplace algebras and of their geometric products in terms of
divided powers of polarization operators. We use these encodings to provide
simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras
and of two typical classes of identities in Cayley-Grassmann algebras
The Dotted straightening algorithm
AbstractIf a homogeneous bracket polynomial is antisymmetric in certain subsets of its points, then it can be represented in an abbreviated form called a dotted bracket expression. These dotted bracket expressions lead to a more compact expression in terms of tableaux than the usual representation. Consequently, we can derive a much more efficient straightening algorithm than the ordinary one for bracket polynomials already given in dotted form. This dotted straightening algorithm gives precisely the same result as the ordinary one, and preserves the dotted property at every step
The role of bideterminants in the representation theory
nuloThis paper provides a brief survey discussing the role of bideterminants in the representation theory
Algebraic aspects of increasing subsequences
We present a number of results relating partial Cauchy-Littlewood sums,
integrals over the compact classical groups, and increasing subsequences of
permutations. These include: integral formulae for the distribution of the
longest increasing subsequence of a random involution with constrained number
of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as
new proofs of old formulae; relations of these expressions to orthogonal
polynomials on the unit circle; and explicit bases for invariant spaces of the
classical groups, together with appropriate generalizations of the
straightening algorithm.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references,
other minor change
A Combinatorial Derivation of the Racah-Speiser Algorithm for Gromov-Witten invariants
Using a finite-dimensional Clifford algebra a new combinatorial product
formula for the small quantum cohomology ring of the complex Grassmannian is
presented. In particular, Gromov-Witten invariants can be expressed through
certain elements in the Clifford algebra, this leads to a q-deformation of the
Racah-Speiser algorithm allowing for their computation in terms of Kostka
numbers. The second main result is a simple and explicit combinatorial formula
for projecting product expansions in the quantum cohomology ring onto the sl(n)
Verlinde algebra. This projection is non-trivial and amounts to an identity
between numbers of rational curves intersecting Schubert varieties and
dimensions of moduli spaces of generalised theta-functions.Comment: 24 pages, 3 figure
Pfaffians and Representations of the Symmetric Group
Pfaffians of matrices with entries z[i,j]/(x\_i+x\_j), or determinants of
matrices with entries z[i,j]/(x\_i-x\_j), where the antisymmetrical
indeterminates z[i,j] satisfy the Pl\"ucker relations, can be identified with a
trace in an irreducible representation of a product of two symmetric groups.
Using Young's orthogonal bases, one can write explicit expressions of such
Pfaffians and determinants, and recover in particular the evaluation of
Pfaffians which appeared in the recent literature.Comment: 28
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