Class Of N-tuples Research Articles

Simultaneous conjugacy classes as combinatorial invariants of finite groups

We consider the problem of counting simultaneous conjugacy classes of n-tuples in a finite group G. Let denote the number of simultaneous conjugacy classes of n-tuples, and the number of simultaneous conjugacy classes of commuting n-tuples in G. We study the generating functions and which are rational functions of t. We show that determines and is completely determined by the class equation of G. We prove that the normalized functions and are invariants of isoclinism families. We compute exponential growth factors of and

Wandering subspaces and inner operators

A noncommutative domain Bφ(H)⊂B(H)n generated by the certain class of n-tuples φ=(φ1,⋯,φn) of formal power series in noncommutative indeterminates Z1,⋯,Zn, admits a universal model MZ=(MZ1,⋯, MZn), acting on a model space H2(φ). In this paper, we provide an explicit description of the wandering subspace W(Q)=Q⊖∑i=1n(MZi⊗IG)Q, where Q is an arbitrary joint invariant subspace under the operators MZ1⊗IG,⋯,MZn⊗IG. Moreover, we also provide a transfer function type characterization of the MZ-inner operators. This extends to the noncommutative setting the corresponding result obtained by Olofsson [7], [8] (when n=1) and by Eschmeier [3] in the multivariable commutative case.

CHAIN EQUIVALENCE ON QUATERNIONS WITH TRIVIAL SUM

We introduce a simple operation on isomorphism classes of n-tuples of quaternions with trivial sum in the Brauer group that corresponds to the notion of Witt's chain equivalence on quadratic forms. The aim of this note is to show that any two classes of n-tuples of quaternions with trivial sum are connected by an iterated application of such simple operations for all n ≤ 6. We also provide some applications to the norm residue homomorphism of degree 2 and to the R-triviality of the corresponding torsors.

Noncommutative Multivariable Operator Theory

In this paper, we study noncommutative domains $${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$$ generated by positive regular free holomorphic functions f and certain classes of n-tuples $${\varphi = (\varphi_1, \ldots, \varphi_n)}$$ of formal power series in noncommutative indeterminates Z 1, . . . , Z n . Noncommutative Poisson transforms are employed to show that each abstract domain $${\mathbb{D}_f^\varphi}$$ has a universal model consisting of multiplication operators (M Z1, . . . , M Z n ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M Z1, . . . , M Z n and show that all pure n-tuples of operators in $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ are compressions of $${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$$ to their coinvariant subspaces. We show that the eigenvectors of $${M_{Z_1}^*, \ldots, M_{Z_n}^*}$$ are precisely the noncommutative Poisson kernels $${\Gamma_\lambda}$$ associated with the elements $${\lambda}$$ of the scalar domain $${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra $${H^\infty(\mathbb{D}_f^\varphi)}$$ . We introduce the characteristic function of an n-tuple $${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$$ , present a model for pure n-tuples of operators in the noncommutative domain $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in $${\mathbb{D}_f^\varphi(\mathcal{H})}$$ .

An Index Formula for an n-Tuple of Shifts on the Polydisk

Let (Mz1, ... , MZn ) be an n-tuple of shift operators on the polydisk 12(Zn); we compress it to a variety of subspaces of 12(Zn) that are combinatorially constructed. The main result is a multivariate Fredholm index formula, which links the indices of the n-tuples to their combinatorial data in the definitions of the subspaces. In [11], Taylor, using homological algebraic methods, introduced a natural spectral theory for n-tuples of commuting operators acting on Banach spaces. Fredholm theory and index theory are developed with respect to this spectral theory [2, 3, 8]. While this several variable spectral theory is a natural generalization of the classical single operator theory (when n = 1), the techniques of computing these spectra are so different from or, much more complicated than, that of single operators. The problem of deciding the spectral pictures of various classes of n-tuples of operators is very essential in multivariate operator theory and has deep applications to many areas of analysis, such as function theory on domains in Cn . Many examples have been investigated through many different techniques [4, 5, 7, 9, 10]. In this note, we use the homological method to compute the spectra and Fredholm indices of a new, interesting class of n-tuples of operators. We will give an index formula for these n-tuples in terms of combinatorial data in their definition. An interesting consequence of this is that some combinatorial property of the operator forces the index to be nonzero. To be more precise, we denote by T = (T1, . .. , Tn) an n-tuple of commuting operators on the Hilbert space H and by K(T, z) = {Kq (T, z), 0q } the Koszul complex of T at z E Cn, where Kq (T, z) is the space of q-cochains and aq is the differential map at degree q. We refer the reader to Taylor's original papers [1 1, 12] for a more detailed account of these concepts. Recall that a point z in Cn is said to be in the Taylor spectrum a(T), if the Koszul Received by the editors October 6, 1992. 1991 Mathematics Subject Classification. Primary 47D25. Research supported in part by a grant from NSF, DMS 90-02969. (? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

A probability measure for character compatibility

It is proved that two undirected binary cladistic characters are compatible iff their smaller states are disjoint or one is a subset of the other. The concept of a cladistic character as an ordered tree of subsets is defined. Cladistic characters that have the same number of elements in their corresponding states are defined to be “nesting equivalent.” The equivalence classes of this relation are called “nestings.” A certain class of n-tuples is shown to have a biunique correspondence with the n!-membered set of all nestings of n binary characters. The model of randomness proposed is that all characters that are nesting equivalent are equally likely. The probability that a pair of undirected binary characters is compatible is derived under this model. This result is extended to collections of undirected binary characters, to collections of directed binary characters, and finally to collections that may include multistate characters. Some proofs are presented which allow a more efficient use of the n-tuple representation of ordered trees of subsets.