Related papers: Loops in SU(2) and Factorization
We consider a refinement of triangular factorization for unitary matrix valued loops.
In previous work we showed that a loop $g\colon S^1 \to {\rm SU}(2)$ has a triangular factorization if and only if the loop $g$ has a root subgroup factorization. In this paper we present generalizations in which the unit disk and its…
In the prequel to this paper, we proved that for a $SU(2,\mathbb C)$ valued loop having the critical degree of smoothness (one half of a derivative in the $L^2$ Sobolev sense), the following statements are equivalent: (1) the Toeplitz and…
We demonstrate a method for general linear optical networks that allows one to factorize any SU($n$) matrix in terms of two SU($n-1)$ blocks coupled by an SU(2) entangling beam splitter. The process can be recursively continued in an…
Previously we showed that a loop in a simply connected compact Lie group K has a unique triangular factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple…
In previous work we proved that for a SU(2,C) valued loop having the critical degree of smoothness (one half of a derivative in the L^2 Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated…
We study in detail the factorization of the newly obtained two-loop four-particle amplitude in superstring theory. In particular some missing factors from the scalar correlators are obtained correctly, in comparing with a previous study of…
We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…
Subsequently to the author's preceding paper, we give full proofs of some explicit formulas about factorizations of $K$-$k$-Schur functions associated with any multiple $k$-rectangles.
We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by C.T.C.Wall. We study this for…
Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…
We recompute the functions describing the collinear factorization of one-loop amplitudes using the unitarity-based method. We present the results in a form suitable for use as an ingredient in two-loop calculations. We also present a…
We show that general $3n-j (n>2)$ symbols of the first kind and the second kind for the group SU(2) can be reformulated in terms of binomial coefficients. The proof is based on the graphical technique established by Yutsis, et al. and…
Root subgroup factorization is a refinement of triangular (or LDU) factorization. For a complex reductive Lie group, and a choice of reduced factorization of the longest Weyl group element, the forward map from root subgroup coordinates to…
We introduce extriangulated factorization systems in extriangulated categories and show that there exists a bijection between $s$-torsion pairs and extriangulated factorization systems. We also consider the gluing of $s$-torsion pairs and…
We discuss the problem of factorisation of the symmetric Macdonald polynomials and present the obtained results for the cases of 2 and 3 variables.
We obtain a multiplication formula for cluster characters on (stably) 2-Calabi-Yau (Frobenius or) triangulated categories. This formula generalizes those known for arbitrary pairs of objects and for Auslander-Reiten triangles. As an…
We make the interprecision transfers explicit in an algorithmic description of iterative refinement and obtain new insights into the algorithm. One example is the classic variant of iterative refinement where the matrix and the…
The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm…
Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. The simple explicit result exhibits connections between group theory,…