Mth Root Of Unity Research Articles

Internal zonotopal algebras and the monomial reflection groups G(m,1,n)

The group G(m,1,n) consists of n-by-n monomial matrices whose entries are mth roots of unity. It is generated by n complex reflections acting on Cn. The reflecting hyperplanes give rise to a (hyperplane) arrangement G⊂Cn. The internal zonotopal algebra of an arrangement is a finite dimensional algebra first studied by Holtz and Ron. Its dimension is the number of bases of the associated matroid with zero internal activity. In this paper we study the structure of the internal zonotopal algebra of the Gale dual of the reflection arrangement of G(m,1,n), as a representation of this group. Our main result is a formula for the top degree component as an induced representation. We also provide results on representation stability, a connection to the Whitehouse representation in type A, and an analog of decreasing trees in type B.

Maps preserving peripheral spectrum of generalized products of operators

Let A1 and A2 be standard operator algebras on complex Banach spaces X1 and X2, respectively. For k⩾2, let (i1,…,im) be a sequence with terms chosen from {1,…,k}, and assume that at least one of the terms in (i1,…,im) appears exactly once. Define the generalized product T1⁎T2⁎⋯⁎Tk=Ti1Ti2⋯Tim on elements in Ai. Let Φ:A1→A2 be a map with the range containing all operators of rank at most two. We show that Φ satisfies that σπ(Φ(A1)⁎⋯⁎Φ(Ak))=σπ(A1⁎⋯⁎Ak) for all A1,…,Ak, where σπ(A) stands for the peripheral spectrum of A, if and only if Φ is an isomorphism or an anti-isomorphism multiplied by an mth root of unity, and the latter case occurs only if the generalized product is quasi-semi Jordan. If X1=H and X2=K are complex Hilbert spaces, we characterize also maps preserving the peripheral spectrum of the skew generalized products, and prove that such maps are of the form A↦cUAU⁎ or A↦cUAtU⁎, where U∈B(H,K) is a unitary operator, c∈{1,−1}.

A Gel'fand Model for the Symmetric Generalized Group

In this article we develop for the symmetric generalized group and K a field of characteristic zero which contains the mth roots of unity, a linear subspace of K[x 1,…, x n ] which is a complex representation of isomorphic to the direct sum of all the irreducible representations of . We also give a parametrization for all the simple -modules, a generator for each simple -module and a formula for the dimensión for all the simple -modules.

The numbers of periodic orbits hidden at fixed points of n-dimensional holomorphic mappings

Let M be a positive integer and let f be a holomorphic mapping from a ball Δn={x∈ℂn:∣x∣<δ} into ℂn such that the origin 0 is an isolated fixed point of both f and fM, the Mth iteration of f. Then one can define the number 𝒪M(f,0), interpreted as the number of periodic orbits of f with period M that are hidden at the fixed point 0. For an n×n matrix A whose eigenvalues are all the same primitive Mth root of unity, we give a sufficient and necessary condition on A such that for any holomorphic mapping f:Δn→ℂn with f(0)=0 and Df(0)=A, if 0 is an isolated fixed point of the Mth iteration fM , then 𝒪M (f,0)≥2.

On efficient balanced codes over the mth roots of unity

Let /spl Phi//sub m//spl sube/ /spl Copf/ be the set of all mth roots of unity, m/spl isin/ IN. A balanced code over /spl Phi//sub m/ is a block code over the alphabet /spl Phi//sub m/ such that each code word is balanced; that is, the complex sum of its components (or weight) is equal to 0. Let B/sub m/(n) be the set of all balanced words of length n over /spl Phi//sub m/. In this correspondence, it is shown that when m is a prime number, the set B/sub m/(n) is not empty if, and only if, m divides n. In this case, the minimum redundancy for a balanced code over /spl Phi//sub m/ of length n is. On the other hand, it is shown that when m=4, the set B/sub 4/(n) is not empty if, and only if, n is even, and in this case, the minimum redundancy for a balanced code over /spl Phi//sub 4/ of length n is. Further, this correspondence completely solves the problem of designing efficient coding methods for balanced codes over /spl Phi//sub m/, when m=4. In fact, it reduces the problem of designing efficient coding schemes for balanced codes over /spl Phi//sub 4/ to the design of efficient balanced codes over the usual bipolar alphabet /spl Phi//sub 2/={-1,+1}.

Pál-type Birkhoff interpolation on nonuniformly distributed points

We study here the regularity of certain Pal-type interpolation problems involving the mth roots of unity along with an additional point ζ and the nth roots of unity. We determine the largest domains for ζ which ensure the regularity of the problems.

Power Bases for 2-Power Cyclotomic Fields

Let ζ be a primitive 2mth root of unity. We prove that Z[α]=Z[ζ] if and only if α=n±ζi for some n, i∈Z, i odd. This is the first example of number fields of arbitrarily large degree for which all power bases for the ring of integers are known.

Exponent Reduction for Radical Abelian Algebras

Let k be a field. A radical abelian algebra over k is a crossed product (K/k,α), where K=k(T) is a radical abelian extension of k, T is a subgroup of K* which is finite modulo k*, and α∈H2(G,K*) is represented by a cocycle with values in T. The main result is that if A is a radical abelian algebra over k, and m=exp(A⊗kk(μ)), where μ denotes the group of all roots of unity, then k contains the mth roots of unity. Applications are given to projective Schur division algebras and projective Schur algebras of nilpotent type.

Torsion of abelian varieties, Weil classes and cyclotomic extensions

Let K⊂C be a field finitely generated over Q, K(a)⊂C the algebraic closure of K and G(K)=Gal (K(a)/K) its Galois group. For each positive integer m we write K(μm) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime [lscr ] we write K([lscr ]) for the subfield of K(a) obtained by adjoining to K all [lscr ]-power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab)⊂K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K=Q then Q(ab)=Q(c) (the Kronecker-Weber theorem). We write χ[lscr ][ratio ]G(K)→Z*[lscr ] for the cyclotomic character defining the Galois action on all [lscr ]-power roots of unity. We write χ[lscr ]= χ[lscr ] mod [lscr ][ratio ]G(K) →Z*[lscr ]→(Z/[lscr ]Z)* for the cyclotomic character defining the Galois action on the [lscr ]th roots of unity. The character χ[lscr ] identifies Gal (K([lscr ])/K) with a subgroup of Z*[lscr ]=Gal (Q([lscr ])/Q). Let μ(Z[lscr ]) be the finite cyclic group μ(Z[lscr ]) of all roots of unity in Z*[lscr ]. Its order is equal to [lscr ]−1 if [lscr ] is odd and 2 if [lscr ]=2. Let Q([lscr ])′ be the subfield of μ(Z[lscr ])-invariants in Q([lscr ]). Clearly, Gal (Q([lscr ])/Q([lscr ])′)=μ(Z[lscr ]) and Gal (Q([lscr ])′/Q)= Z*[lscr ]/μ(Z[lscr ]) is isomorphic to Z[lscr ].

ON PERIODIC REPRESENTATIONS OF QUANTUM GROUPS

We present some results on representations of quantum groups at the root of unity. In the case of SL(2)q, the classification of the finite dimensional irreducible representations is given. For [Formula: see text] with [Formula: see text] a semi-simple or affine Lie algebra and q an mth root of unity (m odd), we classify the representations of dimension [Formula: see text] on which the actions of the Chevalley generators are injective. Finally, we adapt the Gelfand-Zetlin basis to the case of SL(N)q and IU(N)q.

Fusion rules and [formula omitted]-matrix for the composition of regular spins with semi-periodic representations of SL(2) q

We perform the composition, when the parameter q is an mth root of unity, of the spin j representation of SL(2) q with semi-periodic representations characterized by two complex parameters. The result is proved to be completely reducible and it is a direct sum of semi-periodic representations. Denoting by “indecomposable spin representations” the indecomposable representations arising from composition of spin j and spin j' representations, we also prove that the tensor product of indecomposable spin representations with semi-periodic representations is a direct sum of semi-periodic representations. So the set of highest weight representations including spin j representations (1 ⩽ 2 j + 1 ⩽ m), indecomposable spin representations and semi-periodic representations is closed under tensor product. R - matrices which intertwine the tensor products of spin 1 2 representation with semi-periodic representations are derived. These R -matrices satisfy three Yang-Baxter equations.

Periodic representations of SO(5) q

We give explicit expressions for periodic irreducible representations of U q( SO(5)) (or complex quantum De Sitter group) when the parameter of deformation q is an mth root of unity. Their maximal dimension is m 4 when m is odd, as expected, and 1 4 m 4 when m is even.

Irreducible representations of the quantum analogue of SU(2)

We give the complete set of irreducible representations of U(SU(2))q when q is a mth root of unity. In particular, we show that their dimensions are less or equal to m. Some of them are not highest-weight representations.

The brauer group of central separable G-Azumaya algebras

Let R be a commutative ring and G a finite abelian group. In [8], Long developed a Brauer group theory for G-dimodule algebras (i.e., algebras with a compatible G-grading and G-action) and constructed BD(R, G), the Brauer group of G-Azumaya algebras. Within BD(R, G) lies B(R, G), the set of classes of algebras which are R-Azumaya (i.e., central separable) as well as G-Azum$ya. B(R, G) is not always a group; we show that if every cocyle in H2(G, U(R)) is abelian, then it is. When B(R, G) is a group, we call it the Brauer group of central separable G-Azumaya algebras. If R is connected and Pit,(R) = 0 where m is the exponent of G, and if every cocycle in H2(G, U(R)) is abelian, then we show that there is a short exact sequence 1 + (BC(R, G)/B(R)) x (BM(R, G)/B(R)) -+ B(R, G)/B(R) -+ Aut(G) + 1, where B(R) is the usual Brauer group of R, BM(R, G) is the Brauer group of G-module algebras and BC(R, G) is the Brauer group of G-comodule algebras (cf. [S]). If either BM(R, G)/B(R) or BC(R, G)/B(R) is trivial, then the sequence splits. Using the above, we are able to describe BD(Z, G) for any cyclic G, and BD(IW, G) for any cyclic G of odd order. Our sequence provides a generalization of the results [9, Theorem 5.91 and [II, Theorem 4.41 and should be compared to the sequence in [6, Theorem 5.21 obtained under the assumptions that the order of G is a unit in R and R contains a primitive mth root of unity.

A simple proof of an identity of Good

Recently, Good discovered the following curious identity (1). Let θ be a real-valued random variable having cumulants Km ≡ Km(θ) (m = 1, 2, …). Take a set of m statistically independent variables θ(1), θ(2), …, θ(m), each with the same distribution as θ, and form the random variablewhere ω is any primitive mth root of unity (e.g. ω = e2πi/m). Thenwhere 〈.〉 denotes expectation.

The Schur Group of an Algebraic Number Field

If G denotes a finite group and Q the rational field, then the group algebra Q[G] is a direct sum of simple rings, each a full matrix ring over a division ring. One may ask which simple rings arise this way as G is allowed to range over all finite groups. The center of such a simple ring is a field K which is a subfield of Q(sm), where sm is a primitive mth root of unity and m divides the order of G. If we fix our attention on one particular field K, the subset of the Brauer group B(K) of K consisting of all those classes containing an algebra which is isomorphic to a simple summand of Q[G] for some G is a subgroup, S(K), the Schur group of K. This is the same group as the set of classes in B(K) which contain a simple summand of K[G] for some G. We take this as the definition of S(K) when K is any field. Since the elements of B(K) are uniquely determined by Hasse invariants when K is an algebraic number field, it is natural to ask for a description of S(K) in these terms. The main purpose of this paper is to give a description of S(K)p, the p-primary subgroup of S(K), in the case p is odd and K is any abelian extension of Q. In other words we describe up to equivalence in the Brauer group, those simple algebras of odd index which can appear as a direct summand of Q[G]. Some information about the 2-primary part of S(K) is also obtained in the case K contains s, a fourth root of unity. For such a field we determine which numbers can occur as a local index of an element in S(K). Let Q(sm) = L be the least root of unity field containing the given K and let R be the set of rational primes dividing m which ramify between K and L with ramification index divisible by the prime p. Let S(K, R)p denote the subgroup of S(K)p consisting of all elements split at all primes of K which do not divide an element of A. For a prime q not dividing m, let S(K, q)p denote the subgroup of S(K)p of elements split everywhere except at the divisors of q. We obtain the following result.

The generalized Clifford algebra and the unitary group

During the past two years following the first formulation of L-matrix theory [I] the Matscience group has been concerned with the generalised Clifford algebra of matrices which are the mth roots of unity. The generalised algebra was discovered by Yamazaki [2] in 1964 and the matrix representations in the lowest dimension were first given by Morris in 1967 [3]. We shall now present some new results on the subject and point out a surprising and unexpected connection with the generators of the special unitary group. It has been established that there are (2n + 1) matrices L, , L, ,..., L2n+l of dimension mn x mn obeying the two generalised Clifford conditions: