Construction of orthogonal group modules using tableaux

The program of constructing irreducible modules of the classical groups via Young tableaux is extended to include the orthogonal groups O(n) and SO(n). As in Weyl's pioneering work on this topic, each irreducible O(n)-module labelled by a partition λ of l is constructed as a submodule of V _?l. The construction is based on new sets of O(n) and SO(n) standard Young tableaux, closely related to those introduced by Proctor. The reduction of an arbitrary O(n) Young tableau to standard form is accomplished through the use of column antisymmetrisation and Garnir relations, as for GL(n), together with a trace removal process, analogous to that introduced by Bercle for Sp(n). For SO(n) it is necessary to define the associate of a Young tableau to effect the final step of the reduction process. The standardisation procedure is algorithmic and allows matrix representations of the Lie algebras so(n) to be constructed explicitly over the field of rational numbers. It is proved that the matrix elements are rational numbers having only integral powers of 2 in their denominators. All the various steps of the standardisation algorithm are exemplified, as well as the explicit construction of matrices representing elements of so(n). A link between the trace condition on a tableau and a Garnir relation on its associate is established.     

Derivation modules of orthogonal duals of hyperplane arrangements

Let A be an n × d matrix having full rank n. An orthogonal dual A^⊥ of A is a (d-n) × d matrix of rank (d−n) such that every row of A^⊥ is orthogonal (under the usual dot product) to every row of A. We define the orthogonal dual for arrangements by identifying an essential (central) arrangement of d hyperplanes in n-dimensional space with the n × d matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. When n ≥ 5, we show that if the matroid (or the lattice of intersection) of an n-dimensional essential arrangement contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement ^⊥ has projective dimension at least ⌈ n(n+2)/4 ⌉ - 3.Hal Schenck partially supported by NSF DMS 03-11142, NSA MDA 904-03-1-0006, and ATP 010366-0103.     

Quadratic modules and the orthogonal group over polynomial rings

The following conjecture of Parimala is proved: Any quadratic space over a polynomial ring with coefficients from an algebraically closed field of characteristic different from 2 is extended from the coefficient field. In the case of an arbitrary field of characteristic different from 2, an analogous result is obtained for quadratic spaces whose Witt index is at least 2. Also proved are general cancellation theorems for quadratic modules and a stabilization theorem for the orthogonal group over arbitrary polynomial rings.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta, im. V. A. Steklova AN SSSR, Vol. 71, pp. 216–250, 1977.     

Quantum symmetry groups of Hilbert modules equipped with orthogonal filtrations

We define and show the existence of the quantum symmetry group of a Hilbert module equipped with an orthogonal filtration. Our construction unifies the constructions of Banica–Skalski?s quantum symmetry group of a C^?$C^{?}$-algebra equipped with an orthogonal filtration and Goswami?s quantum isometry group of an admissible spectral triple.     

On induced modules and tableaux for the special orthogonal groups

In a recent article, King and Welsh prove a standard basis theorem for the orthogonal groups over a field of characteristic zero utilizing certain tableaux of Proctor that are related to Gelfand patterns. The purpose of the present paper is to extend the previous result to the case of any characteristic other than 2 by working within the context of induced modules for the special orthogonal groups.     

On the Freeness of 3-Arrangements

Hyperplane arrangements in a three-dimensional vector spaceare considered in this paper. A characterization of the freenessof such an arrangement is given in terms of the characteristicpolynomial and a restricted multiarrangement. As an application,the freeness of cones over certain two-dimensional affine arrangementsis proved. 2000 Mathematics Subject Classification 52C35 (primary),32S22 (secondary).     

自由Fisher信息量与合并自由

在算子值非交换概率空间中引入算子值自由Fisher信息量的概念,这一定义是对D.Voiculescu在有迹的von Neumann代数上定义的自由Fisher信息量的推广．证明了算子值自由Fisher信息量与合并自由性是密切相关的,即证明了若干个算子值随机变量的自由Fisher信息量的可加性等价于这些随机变量的合并自由性．并且也类似地得到了Cramer-Rao不等式．     

Freeness of Quasi-Hopf Algebras Over Right Coideal Subalgebras

We will generalize Skryabin's Freeness Theorem [11 Skryabin , S. ( 2007 ). Projectivity and freeness over comodule algebras . Trans. Amer. Math. Soc. 359 : 2597 – 2623 .[Crossref], [Web of Science ®] , [Google Scholar]]to quasi-Hopf algebras. We will show that for a finite dimensional quasi-Hopf algebra H and a right coideal subalgebra K ? H all (H, K)-quasi Hopf bimodules are free K-modules, in particular, H is a free right and left K-module.     

Freeness for 13 lines arrangements is combinatorial

We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane.     

算子值酉随机矩阵及其渐近自由性

该文在算子值非交换概率空间上引入半标准酉随机矩阵的概念, 证明了它是算子值Haar酉元的矩阵模型,并给出了半标准酉随机矩阵的渐近自由判定定理.     

Ramsey and Freeness Properties of Polish Planes

Suppose that X is a Polish space which is not -compact. We provethat for every Borel colouring of X² by countably many colours,there exists a monochromatic rectangle with both sides closedand not -compact. Moreover, every Borel colouring of [X]² byfinitely many colours has a homogeneous set which is closedand not -compact. We also show that every Borel measurable functionf:X² X has a free set which is closed and not -compact. Ascorollaries of the proofs we obtain two results: firstly, theproduct forcing of two copies of superperfect tree forcing doesnot add a Cohen real, and, secondly, it is consistent with ZFCto have a closed subset of the Baire space which is not -compactand has the property that, for any three of its elements, noneof them is constructible from the other two. A similar proofshows that it is consistent to have a Laver tree such that noneof its branches is constructible from any other branch. Thelast four results answer questions of Goldstern and Brendle.2000 Mathematics Subject Classification: 03E15, 26B99, 54H05.     

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Freeness of linear and quadratic forms in von Neumann algebras

We characterize the semicircular distribution by freeness of linear and quadratic forms in noncommutative random variables from tracial W^?-probability spaces with relaxed moment conditions.