scholarly journals The Equations Defining Affine Grassmannians in Type A and a Conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

2020 ◽  
Author(s):  
Dinakar Muthiah ◽  
Alex Weekes ◽  
Oded Yacobi
Keyword(s):  
Symmetric Functions ◽  
Fock Space ◽  
Linear Part ◽  
Nilpotent Operator ◽  
Affine Grassmannian ◽  
Sato Grassmannian ◽  
Affine Grassmannians ◽  
Finite Dimensional ◽  
Plücker Relations ◽  
Plücker Embedding

Abstract The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called shuffles, and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps; first, we reinterpret the shuffle equations in terms of Frobenius twists of symmetric functions. Using this, we reduce to a finite-dimensional problem, which we solve. For the 2nd step, we introduce a finite-dimensional analogue of the affine Grassmannian of $SL_n$, which we conjecture to be precisely the reduced subscheme of a finite-dimensional Grassmannian consisting of subspaces invariant under a nilpotent operator.

scholarly journals Peterson Isomorphism in K-theory and Relativistic Toda Lattice

2018 ◽  
Vol 2020 (19) ◽  
pp. 6421-6462 ◽  
Author(s):  
Takeshi Ikeda ◽  
Shinsuke Iwao ◽  
Toshiaki Maeno
Keyword(s):  
Symmetric Functions ◽  
Toda Lattice ◽  
Flag Variety ◽  
The Other ◽  
Initial Condition ◽  
Birational Morphism ◽  
Affine Grassmannian ◽  
Other Hand ◽  
K Theory ◽  
Relativistic Toda Lattice

Abstract The K-homology ring of the affine Grassmannian of $SL_{n}(\mathbb{C})$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum K-theory of the flag variety $F\,\! l_{n}$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars’s relativistic Toda lattice with unipotent initial condition. From this result, we obtain a K-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart–Maeno’s quantum Grothendieck polynomial associated with a Grassmannian permutation.

scholarly journals Semi-infinite Plücker Relations and Weyl Modules

2018 ◽  
Vol 2020 (14) ◽  
pp. 4357-4394 ◽  
Author(s):  
Evgeny Feigin ◽  
Ievgen Makedonskyi
Keyword(s):  
Type A ◽  
Character Formula ◽  
Flag Varieties ◽  
Coordinate Ring ◽  
Weyl Modules ◽  
Plücker Relations ◽  
Plücker Embedding ◽  
Formal Version ◽  
The Ideal

Abstract The goal of this paper is two-fold. First, we write down the semi-infinite Plücker relations, describing the Drinfeld–Plücker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, that is, the quotient by the ideal generated by the semi-infinite Plücker relations. We establish the isomorphism with the algebra of dual global Weyl modules and derive a new character formula.

The Hodge Operator in Fermionic Fock Space

2005 ◽  
Vol 70 (7) ◽  
pp. 979-1016 ◽  
Author(s):  
Leszek Z. Stolarczyk
Keyword(s):  
Fock Space ◽  
Differential Forms ◽  
Grassmann Algebra ◽  
Basis Set ◽  
Electron Systems ◽  
Linear Transformations ◽  
Spin Orbital ◽  
Creation And Annihilation Operators ◽  
Finite Dimensional ◽  
Hodge Operator

The Hodge operator ("star" operator) plays an important role in the theory of differential forms, where it serves as a tool for the switching between the exterior derivative and co-derivative. In the theory of many-electron systems involving a finite-dimensional fermionic Fock space, one can define the Hodge operator as a unique (i.e., invariant with respect to linear transformations of the spin-orbital basis set) antilinear operator. The similarity transformation based on the Hodge operator results in the switching between the fermion creation and annihilation operators. The present paper gives a self-contained account on the algebraic structures which are necessary for the construction of the Hodge operator: the fermionic Fock space, the corresponding Grassmann algebra, and the generalized creation and annihilation operators. The Hodge operator is then defined, and its properties are reviewed. It is shown how the notion of the Hodge operator can be employed in a construction of the electronic time-reversal operator.

scholarly journals Holomorphic Fock spaces for positive linear transformations

2006 ◽  
Vol 98 (2) ◽  
pp. 262 ◽  
Author(s):  
R. Fabec ◽  
G. Ólafsson ◽  
A.N. Sengupta
Keyword(s):  
Gaussian Measure ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Inner Product ◽  
Fock Spaces ◽  
Linear Transformations ◽  
Positive Real ◽  
Bargmann Transform ◽  
Finite Dimensional ◽  
Real Linear

Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt{\det A}} {\pi^n}e^{-\Re\langle Az,z\rangle}\,dz$ is described in terms of the linear and antilinear decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal-Bargmann transform, which we also study in the Gaussian-measure setting.

Unbounded Perturbations of Boson Equilibrium States in Fock Space

1991 ◽  
Vol 46 (4) ◽  
pp. 293-303
Author(s):  
Reinhard Honegger
Keyword(s):  
Fock Space ◽  
Relativistic Quantum ◽  
Trace Class ◽  
Coupling Term ◽  
Relativistic Quantum Theory ◽  
Interaction Operator ◽  
Gas Radiation ◽  
Finite Dimensional ◽  
Interacting Systems ◽  
Photon Interactions

Abstract Based on the linear coupling term in the non-relativistic quantum theory of matter-photon interactions, the coupling of a finite dimensional quantum system (finitely many finite-level atoms) with the boson gas (radiation field) in thermal equilibrium by means of perturbation theoretical methods is calculated. For the perturbation term of the Dyson expansion the unbounded interaction operator is taken. By a detailed analysis of the perturbation integrals and series, it is possible to derive the trace-class property of e-ßH , where H is the corresponding hamiltonian of the interacting systems and ß the inverse temperature

scholarly journals W-ALGEBRAS FROM HEISENBERG CATEGORIES

2016 ◽  
Vol 17 (5) ◽  
pp. 981-1017 ◽  
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Anthony M. Licata ◽  
Joshua Sussan
Keyword(s):  
Symmetric Functions ◽  
Fock Space ◽  
Hochschild Homology ◽  
Space Representation

The trace (or zeroth Hochschild homology) of Khovanov’s Heisenberg category is identified with a quotient of the algebra$W_{1+\infty }$. This induces an action of$W_{1+\infty }$on the center of the categorified Fock space representation, which can be identified with the action of$W_{1+\infty }$on symmetric functions.

scholarly journals Real parts of quasi-nilpotent operators

1979 ◽  
Vol 22 (3) ◽  
pp. 263-269 ◽  
Author(s):  
P. A. Fillmore ◽  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Dimensional Case ◽  
Nilpotent Operator ◽  
The Real ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Nilpotent Operators ◽  
Finite Dimensional Case ◽  
Adjoint Operators

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

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