scholarly journals Equivariant Giambelli formula for the symplectic Grassmannians — Pfaffian Sum Formula

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Takeshi Ikeda ◽  
Tomoo Matsumura
Keyword(s):  
Vector Space ◽  
Closed Formula ◽  
International Audience ◽  
Sum Formula ◽  
Isotropic Subspaces ◽  
Espace Vectoriel ◽  
Symplectic Vector Space ◽  
Symplectic Grassmannians ◽  
Schubert Class

International audience We prove an explicit closed formula, written as a sum of Pfaffians, which describes each equivariant Schubert class for the Grassmannian of isotropic subspaces in a symplectic vector space On démontre une formule close explicite, écrite comme une somme de Pfaffiens, qui décrit toute classe de Schubert équivariante pour la Grassmannienne des sous-espaces isotropes dans un espace vectoriel symplectique.

scholarly journals Generalized Polarization Modules (extended abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Héctor Blandin
Keyword(s):  
Vector Space ◽  
Vandermonde Determinant ◽  
Homogeneous Polynomials ◽  
Generalized Polarization ◽  
Partial Derivatives ◽  
Diagonal Harmonics ◽  
The Family ◽  
International Audience ◽  
Stable Family ◽  
Espace Vectoriel

International audience This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\ell$ sets of $n$ variables $x_{ij}$ with $1\le i \le \ell$ and $1 \le j \le n$. Given a $\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_{ij}$ the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains $F$ is the polarization module generated by the family $F$. These polarization modules are all representation of the direct product $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. In order to study the decomposition into irreducible submodules, we compute the graded Frobenius characteristic of these modules. For several cases of $\mathfrak{S}_n$-stable families of homogeneous polynomials in n variables, for every $n \ge 1$, we show general formulas for this graded characteristic in a global manner, independent of the value of $\ell$. Ce travail s'inscrit dans la lignée de recherche des travaux de M. Haiman sur le théorème de l'opérateur (ex-conjecture de l'opérateur). Ce théorème affirme que le plus petit $\mathfrak{S}_n$-module clos par dérivation partielle et clos par l'action des opérateurs de polarisation qui contient le déterminant de Vandermonde est l'espace des polynômes harmoniques diagonaux. On commence par généraliser le contexte du théorème de l'opérateur au contexte de polynômes à ensembles de $n$ variables $x_{ij}$ avec $1\le i \le \ell$ et $1 \le j \le n$. Étant donnée une famille $\mathfrak{S}_n$-stable $F$ des polynômes homogènes en les variables $x_{ij}$, le plus petit espace vectoriel $\mathcal{M}_F$ clos par dérivation partielle et clos par léaction des opérateurs de polarisation contenant $F$ est le module de polarisation engendré par la famille $F$. Les modules $\mathcal{M}_F$ sont tous des représentations du produit direct $\mathfrak{S}_n \times GL_\ell(\mathbb{C})$. Dans le but d'étudier la décomposition en sous-modules irréductibles on calcule la caractéristique de Frobenius graduée de ces modules. Pour plusieurs cas de familles homogènes $\mathfrak{S}_n$-stables constituées des polynômes homogènes à $n$ variables, pour tout $n \ge 1$, on démontre des formules générales pour cette caractéristique graduée de façon globale, indépendante de la valeur de $\ell$.

scholarly journals Isotropical Linear Spaces and Valuated Delta-Matroids

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Felipe Rincón
Keyword(s):  
Symmetric Matrix ◽  
Combinatorial Theory ◽  
Linear Spaces ◽  
Dimensional Vector Space ◽  
Type D ◽  
Matroid Polytopes ◽  
International Audience ◽  
Isotropic Subspaces ◽  
Tropical Basis ◽  
Espace Vectoriel

International audience The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an $n \times n$ skew-symmetric matrix. Its points correspond to $n$-dimensional isotropic subspaces of a $2n$-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type $D$. La variété spinorielle est decoupée par les relations quadratiques de Wick parmi les Pfaffiens principaux d'une matrice antisymétrique $n \times n$. Ses points correspondent aux sous-espaces isotropes à $n$ dimensions d'un espace vectoriel de dimension $2n$. Dans cet article nous tropicalisons cette description, et nous développons une théorie combinatoire de vecteurs tropicaux de Wick et d'espaces linéaires tropicaux qui sont tropicalement isotropes. Nous caractérisons des vecteurs tropicaux de Wick en termes de subdivisions des polytopes Delta-matroïde, et nous étudions dans quelle mesure les relations de Wick forment une base tropicale. Notre théorie généralise plusieurs résultats pour les espaces linéaires tropicaux et évaluait des matroïdes à la classe des matroïdes de Coxeter du type $D$.

scholarly journals Isomorphic exponential Weyl algebras

1991 ◽  
Vol 33 (1) ◽  
pp. 7-10 ◽  
Author(s):  
P. L. Robinson
Keyword(s):  
Vector Space ◽  
Weyl Algebra ◽  
Associative Algebras ◽  
Exponential Form ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Weyl Algebras ◽  
The Subject ◽  
Symplectic Vector Space

Canonically associated to a real symplectic vector space are several associative algebras. The Weyl algebra (generated by the Heisenberg commutation relations) has been the subject of much study; see [1] for example. The exponential Weyl algebra (generated by the canonical commutation relations in exponential form) has been less well studied; see [8].

scholarly journals Quaternionic Grassmannians and Borel classes in algebraic geometry

2021 ◽  
Vol 33 (1) ◽  
pp. 97-140
Author(s):  
I. Panin ◽  
C. Walter
Keyword(s):  
Algebraic Geometry ◽  
Symplectic Form ◽  
Cell Structure ◽  
Content Type ◽  
Open Subscheme ◽  
Oriented Cohomology ◽  
Symplectic Cobordism ◽  
Sum Formula ◽  
Symplectic Vector Space ◽  
Witt Groups

The quaternionic Grassmannian H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) is the affine open subscheme of the usual Grassmannian parametrizing those 2 r 2r -dimensional subspaces of a 2 n 2n -dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular, we have HP n = H Gr ⁡ ( 1 , n + 1 ) \operatorname {HP}^n = \operatorname {H Gr}(1,n+1) . For a symplectically oriented cohomology theory A A , including oriented theories but also the Hermitian K \operatorname {K} -theory, Witt groups, and algebraic symplectic cobordism, we have A ( HP n ) = A ( pt ) [ p ] / ( p n + 1 ) A(\operatorname {HP}^n) = A(\operatorname {pt})[p]/(p^{n+1}) . Borel classes for symplectic bundles are introduced in the paper. They satisfy the splitting principle and the Cartan sum formula, and they are used to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes. The cell structure of the H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) exists in cohomology, but it is difficult to see more than part of it geometrically. An exception is HP n \operatorname {HP}^n where the cell of codimension  2 i 2i is a quasi-affine quotient of A 4 n − 2 i + 1 \mathbb {A}^{4n-2i+1} by a nonlinear action of G a \mathbb {G}_a .

scholarly journals On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

scholarly journals Matrix product and sum rule for Macdonald polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Luigi Cantini ◽  
Jan De Gier ◽  
Michael Wheeler
Keyword(s):  
Exclusion Process ◽  
Macdonald Polynomials ◽  
Stationary Measure ◽  
Asymmetric Exclusion Process ◽  
Matrix Product ◽  
Macdonald Polynomial ◽  
Sum Rule ◽  
Infinite Dimensional ◽  
International Audience ◽  
Sum Formula

International audience We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one.

scholarly journals Pfaffian sum formula for the symplectic Grassmannians

2015 ◽  
Vol 280 (1-2) ◽  
pp. 269-306 ◽  
Author(s):  
Takeshi Ikeda ◽  
Tomoo Matsumura
Keyword(s):  
Sum Formula ◽  
Symplectic Grassmannians
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