1,277 research outputs found

### About Division by 1

The Euclidean division of two formal series in one variable produces a
sequence of series that we obtain explicitly, remarking that the case where one
of the two initial series is 1 is sufficiently generic. As an application, we
define a Wronskian of symmetric functions.Comment: 7 page

### The 6 Vertex Model and Schubert Polynomials

We enumerate staircases with fixed left and right columns. These objects
correspond to ice-configurations, or alternating sign matrices, with fixed top
and bottom parts. The resulting partition functions are equal, up to a
normalization factor, to some Schubert polynomials.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA

### The Hecke algebra and structure constants of the ring of symmetric polynomials

We give half a dozen bases of the Hecke algebra of the symmetric group, and
relate them to the basis of Geck-Rouquier, and to the basis of Jones, using
matrices of change of bases of the ring of symmetric polynomials

### The differential equation satisfied by a plane curve of degree n

Eliminating the arbitrary coefficients in the equation of a generic plane
curve of order $n$ by computing sufficiently many derivatives, one obtains a
differential equation. This is a projective invariant. The first one,
corresponding to conics, has been obtained by Monge. Sylvester, Halphen, Cartan
used invariants of higher order. The expression of these invariants is rather
complicated, but becomes much simpler when interpreted in terms of symmetric
functions

### 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

### Adding \pm 1 to the argument of an Hall-Littlewood polynomial

Shifting by \pm 1 powers sums: p_i \to p_i \pm 1 induces a transformation on
symmetric functions that we detail in the case of Hall-Littlewood polynomials.
By iteration, this gives a description of these polynomials in terms of plane
partitions, as well as some generating functions. We recover in particular an
identity of Warnaar related to Rogers-Ramanujan identities

### Generalisation of Scott permanent identity

Scott considered the determinant of 1/(y-z)^2, with y,z running over two sets
X,Y of size n, and determined its specialisation when Y and Z are the roots of
y^n-a and z^n-b. We give the same specialisation for the determinant
1/\prod_x(xy-z), where {x} is an arbitrary set of indeterminates. The case of
the Gaudin-Izergin-Korepin is for {x}={q,1/q}.Comment: 5 page

### Crystal Graphs and $q$-Analogues of Weight Multiplicities for the Root System $A_n$

We give an expression of the $q$-analogues of the multiplicities of weights
in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal
graph attached to the corresponding U_q(\sl_{n+1})-modules. As an
application, we describe multivariate polynomial analogues of the
multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to
appear in Lett. Math. Phy

### Orthogonal divided differences and Schubert polynomials, \~P-functions, and vertex operators

We give several new formulas which are useful for Schubert Calculus
associated with the orthogonal groups and related orthogonal degeneracy loci.Comment: 26 pages, AMSTEX. to appear in volume 48 of the Michigan Math.J. in
honor of William Fulto

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