量子MacMahon主定理的特殊化和推广

The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}(m|n)$

We prove an analogue of the MacMahon Master Theorem for the right quantum superalgebras. In particular, we obtain a new and simple proof of this theorem for the right quantum algebras. In the super case the theorem is then used to construct higher order Sugawara operators for the affine Lie superalgebra $\widehat{\mathfrak{gl}}(m|n)$ in an explicit form. The operators are elements of a completed universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$ at the critical level. They occur as the coefficients in the expansion of a noncommutative Berezinian and as the traces of powers of gener…

Extended MacMahon-Schwinger's Master Theorem and Conformal Wavelets in Complex Minkowski Space

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L^2_h(D_4,dν_λ) and L^2_h(C^4_+,d\tildeν_λ) of square integrable h…

Foata基本变换的推广及其在右量子代数中的应用

A generalization of Foata's fundamental transformation and its applications to the right-quantum algebra

右-量子代数是Garoufalten，Lê和Zeilberger在MacMahon主定理的量子推广中引入的.由于Konvalinka和Pak对这个恒等式的组合证明，以及最近对右量子西尔维斯特行列式恒等式的证明，大量使用了与Foata引入的第一个基本变换有关的双射。本文明确了这种变换和右量子线性代数恒等式之间的联系，应用包括右量子矩阵逆定理的一个新的组合证明，以及两个新的结果，右量子Jacobi比定理和右量子MacMahon主定理的推广.

Quantum-inspired permanent identities

The permanent is pivotal to both complexity theory and combinatorics. In quantum computing, the permanent appears in the expression of output amplitudes of linear optical computations, such as in the Boson Sampling model. Taking advantage of this connection, we give quantum-inspired proofs of many existing as well as new remarkable permanent identities. Most notably, we give a quantum-inspired proof of the MacMahon master theorem as well as proofs for new generalizations of this theorem. Previous proofs of this theorem used completely different ideas. Beyond their purely combinatorial applicat…