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Register a new userThe Bloch-Okounkov correlation functions, a classical half-integral case
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David G. Taylor
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Bloch and Okounkovs correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of $hgl_infty$-modules of level one. Recent works have calculated these character functions for higher levels for $hgl_infty$ and its Lie subalgebras of classical type. Here we obtain these functions for the subalgebra of type $D$ of half-integral levels and as a byproduct, obtain $q$-dimension formulas for integral modules of type $D$ at half-integral level.
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Bloch and Okounkov introduced an $n$-point correlation function on the fermionic Fock space and found a closed formula in terms of theta functions. This function affords several distinguished interpretations and in particular can be formulated as correlation functions on irreducible $hat{gl}_infty$-modules of level one. These correlation functions have been generalized for irreducible integrable modules of $hat{gl}_infty$ and its classical Lie subalgebras of positive levels by the authors. In this paper we extend further these results and compute the correlation functions as well as the $q$-dimensions for modules of $hat{gl}_infty$ and its classical subalgebras at negative levels.
Let $Z$ be the symmetric cone of $r times r$ positive definite Hermitian matrices over a real division algebra $mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_lambda$ -- indexed by partitions $lambda$ of length at most $r$, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration $Y longleftarrow X longrightarrow Z$ where $Y$ is the Grassmanian of $r$-dimensional subspaces of $mathbb F^n $ with $n geq 2r$. Using this we construct a family of invariant differential operators $D_{lambda,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{lambda,s}$ are given by specializations of Okounkov interpolation polynomials.
The detailed derivation of the quantum Landau-Lifshitz-Bloch (qLLB) equation for simple spin-flip scattering mechanisms based on spin-phonon and spin-electron interactions is presented and the approximations are discussed. The qLLB equation is written in the form, suitable for comparison with its classical counterpart. The temperature dependence of the macroscopic relaxation rates is discussed for both mechanisms. It is demonstrated that the magnetization dynamics is slower in the quantum case than in the classical one.
We show that quite universally the holonomicity of the complexity function of a big divisor on a projective variety does not predict the polyhedrality of the Newton-Okounkov body associated to every flag.
Classical oscillator differential equation is replaced by the corresponding (finite time) difference equation. The equation is, then, symmetrized so that it remains invariant under the change d going to -d, where d is the smallest span of time. This symmetric equation has solutions, which come in reciprocally related pairs. One member of a pair agrees with the classical solution and the other is an oscillating solution and does not converge to a limit as d goes to 0. This solution contributes to oscillator energy a term which is a multiple of half-integers.
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