Matrices In GL Research Articles

The conjugacy problem in GL (n,Z)

We present a new algorithm that, given two matrices in GL ( n , Q ) , decides if they are conjugate in GL ( n , Z ) and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in GL ( n , Z ) of a matrix in GL ( n , Q ) . We do this by reducing these problems, respectively, to the isomorphism and automorphism group problems for certain modules over rings of the form O K [ y ] / ( y l ) , where O K is the maximal order of an algebraic number field and l ∈ N , and then provide algorithms to solve the latter. The algorithms are practical and our implementations are publicly available in Magma.

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2

We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various p-adic analytic and adèlic profinite groups of type A 2 . This has consequences for the representation zeta functions of arithmetic groups Γ ⊂ H ( k ) , where k is a number field and H is a k-form of SL 3 : assuming that Γ possesses the strong congruence subgroup property, we obtain precise, uniform estimates for the representation growth of Γ. Our results are based on explicit, uniform formulae for the representation zeta functions of the p-adic analytic groups SL 3 ( o ) and SU 3 ( o ) , where o is a compact discrete valuation ring of characteristic 0. These formulae build on our classification of similarity classes of integral p-adic 3 × 3 matrices in gl 3 ( o ) and gu 3 ( o ) , where o is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions. Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form SL 3 ( o ) , SU 3 ( o ) , GL 3 ( o ) , and GU 3 ( o ) , arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of o is either 0 or sufficiently large. Analysis of some of these formulae leads us to observe p-adic analogues of ‘Ennola duality’.

On Representations of Quantum Conjugacy Classes of GL(n)

Let $O$ be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by $T$ the maximal torus of diagonal matrices in GL(n). With every $a\in O\cap T$ we associate a highest weight module $M_a$ over the quantum group $U_q(gl(n))$ and an equivariant quantization $C_{h,a}[O]$ of the polynomial ring $C[O]$ realized by operators on $M_a$. All quantizations $C_{h,a}[O]$ are isomorphic and can be regarded as different exact representations of the same algebra, $C_{h}[O]$. Similar results are obtained for semisimple adjoint orbits in $gl(n)$ equipped with the canonical GL(n)-invariant Poisson structure.

Some complete intersection symplectic quotients in positive characteristic: Invariants of a vector and a covector

Given a linear action of a group G on a K-vector space V, we consider the invariant ring K [ V ⊕ V ⁎ ] G , where V ⁎ is the dual space. We are particularly interested in the case where V = F q n and G is the group U n of all upper unipotent matrices or the group B n of all upper triangular matrices in GL n ( F q ) . In fact, we determine F q [ V ⊕ V ⁎ ] G for G = U n and G = B n . The result is a complete intersection for all values of n and q. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of “doubly parametrized” series of group actions whose invariant rings are known to have a uniform description.

Minimal affine coordinates for [formula omitted] character varieties of free groups

Let X r be the moduli of SL ( 3 , C ) representations of a rank r free group. In this paper we determine minimal generators of the coordinate ring of X r . This at once gives explicit global coordinates for the moduli and determines the dimension of the moduli's minimal affine embedding. Along the way, we utilize results concerning the moduli of r-tuples of matrices in gl ( 3 , C ) . Consequently, we also state general invariant theoretic correspondences between the coordinate rings of the moduli of r-tuples of elements in gl ( n , C ) , sl ( n , C ) , and SL ( n , C ) .

Unipotent Conjugacy in General Linear Groups

ABSTRACT Let U(n,q) be the group of upper uni-triangular matrices in GL(n,q), the n-dimensional general linear group over the field of q elements. The number of U(n,q)-conjugacy classes in GL(n,q) is, as a function of q, for fixed n, a polynomial in q with integral coefficients.

On the asymptotic number of non-equivalent q-ary linear codes

Let M n , q ⊂ GL ( n , F q ) be the group of monomial matrices, i.e., the group generated by all permutation matrices and diagonal matrices in GL ( n , F q ) . The group M n , q acts on the set V ( F q n ) of all subspaces of F q n . The number of orbits of this action, denoted by N n , q , is the number of non-equivalent linear codes in F q n . It was conjectured by Lax that N n , q ∼ | V ( F q n ) | n ! ( q - 1 ) n - 1 as n → ∞ . We confirm this conjecture in this paper.

A characterization of stable and Borel ideals

In this paper a combinatorial characterization of stable, lex-segment and Borel ideals in P(n) := k[X1, ..., Xn] (assuming chark = 0) is given. The last ones are special monomial ideals, widely studied after the fundamental result of A. Galligo [7] (chark = 0) and D. Bayer–M. Stillman [1] (any characteristic) stating that for every homogeneous ideal a ⊆ P(n) and generic change of coordinates the initial ideal does not vary and is actually fixed by the Borel subgroup of triangular matrices in Gl(n,k). Such a monomial ideal is called generic initial ideal of a and denoted gin (a). Also the first ones are monomial ideals (including the Borel ones) studied after S. Eliahou–M. Kervaire [6] that gave a handy way of computing their minimal free resolutions. We want to characterize stable, Borel and lex-segment ideals a ⊆ P(n) by means of the associated order ideal N (a) ⊆ N, corresponding to the terms outside a. We note first that N (a) reflects the geometric structure of the reduced union of linear varieties X(a) ⊆ P associated to a ⊆ P(n), via the well-known lifting-construction due to Hartshorne (see [13]). To each monomial ideal a ⊆ P(n) we associate a numerical character, denoted e(a) and called e-vector of a, consisting of a sequence of symbols ∞ and positive integers intercalated by finite groups of /’s. We notice that symbols ∞ do not occur if a is 0-dimensional. The numerical character e(a) gives in a very compact way the ‘geometric configuration’ of the order ideal N (a) ⊆ N and a canonical system of generators of a that in the stable case is precisely the minimal system G(a) of monomial generators of a (see Thm. 3.6). Namely, assuming a ⊆ P(n) stable, let G(a) = {m1, . . . , mr} and let m1 <L m2 <L · · · <L mr , where <L is the lexicographic ordering, then the entries of e(a) belonging to N are ordinately

Applications of Bruhat decompositions to complex hyperbolic geometry

The double coset space AΛ (n, ℂ) / U (n − 1, 1) is studied, where A consists of the diagonal matrices in GL (n, ℂ). This space naturally arises in the harmonic analysis on the hermitian symmetric space GL (n, ℂ) / U (n − 1, 1). It is shown here that these double cosets also represent a class of basic invariants related to complex hyperbolic geometry. An algebraic parametrization for the double cosets is given and it is shown how this may be used to conveniently compute the geometric invariants.

A tensor product action on q-clan generallzed quadrangles with q = 2 e

The Fundamental Theorem of q-clan geometry implies (among other things) that all automorphisms of the generalized quadrangle GQ( C ) of order ( q 2, q) associated with a q-clan C which fix a special pair of points are automorphisms of the elation group of the quadrangle. When q = 2 e , with a slight modification of the usual representation, we describe these automorphisms in terms of tensor products of pairs of matrices in GL(2, q). The resulting efficiency in computation allows a simplified description of the automorphisms of GQ( C ). We apply the general theory to give an improve description of the induced stabilizers of the ovals in PG(2, q) that are associated with the new Subiaco q-clans introduced and studied in the recent literature.

A Note on the Unit Group of ZS 3

The group V( ZS3) of units of augmentation 1 in ZS3 is characterized as the group of all doubly stochastic matrices in GL(3, Z). Two normal complements are presented, one of which is torsion free while the other contains elements of order 2. Hughes and Pearson [3] characterized the group V = V(ZS3) of units of argmentation 1 in ZS3 by showing that V is isomorphic to the subgroup of GL(2, Z) consisting of matrices whose column sums are 1 modulo 3. Their approach was to construct a 6 X 6 matrix from a full set of irreducible representations of S3, invert the matrix, then solve a system of six linear congruences modulo 6. The same approach, with slight modifications, was used subsequently by Polcino Milies [4] to describe units in ZD4 and by the authors [1] to describe units in ZA4. In this paper we use a different method to obtain a new description of V(ZS3) as the group of all doubly stochastic matrices in GL(3, Z). Working in GL(3, Z) instead of GL(2, Z) permits us to exploit the fact that a convex combination of permutation matrices is always doubly stochastic; it is not necessary to invert a 6 x 6 matrix or to solve a system of linear congruences. It is well known that S3 has a torsion free normal complement in V(ZS3). (This is a special case of a theorem of Passman and Smith [6] and of a more general theorem of Cliff, Sehgal, and Weiss [2].) On the other hand, it is easy to see that the doubly stochastic matrices which are the identity modulo 2 are a normal complement to S3 in our presentation, and that this complement contains elements of order 2. We represent S3 = Ka,b la2 = b= 1 and a-lba = b2) by 0 1 0 0 1 0 p(a)=A=1 10 0 p(b) = B =0 0 1 ?0 O l1 0 0 and extend p linearly to ZS3. Whenever a = Ecija'bJ E ZS3, it is clear that Coo + C12, col + clo, C02 + Cl1 (1) P(a)= C02 + Clo ,coo + C11, Co0 + C12 Coi + Cll, C02 + C12, Coo + C10 In addition, if a is a unit of augmentation 1, then p(a) is clearly a doubly stochastic matrix in GL(3, Z). Received by the editors December 9, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 20C05: Secondary 20C10. ?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

Homology of integral upper-triangular matrices

We calculate the homology of the multiplicative group of integral upper-triangular n × n n \times n matrices at all primes p ⩾ n − 1 p \geqslant n - 1 .

Detecting Products of Elementary Matrices in GL 2 ( Z [ √d

Detecting Products of Elementary Matrices in GL 2 ( Z [ √d

Commutators of Matrices with Prescribed Determinant

Let K be a commutative field, let GL(n, K) be the multiplicative group of all non-singular n × n matrices with elements from K, and let SL(n, K) be the subgroup of GL(n, K) consisting of all matrices in GL(n, K) with determinant one. We denote the determinant of matrix A by |A|, the identity matrix by In, the companion matrix of polynomial p(λ) by C(p(λ)), and the transpose of A by AT. The multiplicative group of nonzero elements in K is denoted by K*. We let GF(pn) denote the finite field having pn elements.