Full Linear Group Research Articles

Linear groups associated with elements of a group algebra

Let σ → c(σ) be an arbitrary nonzero function from Sm into a field K. If A = [Aij] is an m-square matrix over K, define dc(A)=∑σ∈Smc(σ)Πi=1m Aiσ(i) We are concerned with the characterization of the group of (nonsingular) matrices A satisfying dc(AX) = dc(X) for all m-square matrices X over K. The author previously defined [13] a Galois connection between the set Λ of left ideals of the group algebra K[Sm], ordered by inclusion, and the set Γ of subgroups of the full linear group, also ordered by inclusion. We characterize the closed elements of Λ by the closure operation arising from this connection.

Subgroups of the full linear group over a semilocal ring

Let Λ be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealm⊂C contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, Λ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals σ such that G(σ)⩽H⩽N(σ), whereN(σ) is the normalizer of the D-net subgroup σ. A transparent classification of subgroups GL(n, Λ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN(σ)/G(σ) is studied.

Contractibility of the full general linear group of thel 2(A)

Contractibility of the full general linear group of thel 2(A)

Would leibniz lie to you? (three aspects of the affine connection)

Second derivatives, even in ordinary calculus, have a certain unreality since they can’t be invariantly defined. Making them invariant, the dream of Leibniz, is equivalent to assuming an affine connection, which has the drawback that a natural unique one cannot generally be found. Thus, one is led to the conviction that second derivatives by themselves aren’t meaningful, that only the composite object consisting of first and second derivative together has significance. This object and its dual, acceleration with velocity squared, satisfy generalized chain rules which lead to an idea of tensor with invariance group wider than the full linear group of Jacobian matrices. In this broader world, who knows how bright the strange birds will sing?

On the decomposition numbers of the finite general linear groups

Let G = GL,,(q), q a prime power, and let r be an odd prime not dividing q. Let s be a semisimple element of G of order prime to r and assume that r divides qdeg(X\) 1 for all elementary divisors A of s. Relating representations of certain Hecke algebras over symmetric groups with those of G, we derive a full classification of all modular irreducible modules in the r-block Bh of G with semisimple part s. The decomposition matrix D of Bs may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the Hecke algebras above. Moreover D is lower unitriangular. This applies in particular to all r-blocks of G if r divides q 1. Thus, in this case, the r-decomposition matrix of G is lower unitriangular. Introduction. The modular representation theory of finite groups of Lie type defined over a field of characteristic p is naturally divided into the cases of equal characteristic r = p and unequal characteristic r / p. This paper begins the study of decomposition numbers in the unequal characteristic case of the finite general linear groups when r > 2. Let G be the full linear group of degree n over GF(q) and 2 /= r be a prime not dividing q. Let s c G be semisimple and consider the geometric conjugacy class (s)G of irreducible characters corresponding to the G-conjugacy class of s. Using basic properties of the Deligne-Lusztig operators introduced in [7] we will determine a parabolic subgroup P = Ps of G and a cuspidal irreducible character of X of P such that (s)G is just the set of constituents of XG, the induced character. Let { R, R, K } be an r-modular system for G and M a KP-module affording X. Then End KG(M ) is called the Hecke algebra of M (compare [5]) and has been calculated in a more general context by R. B. Howlett and G. I. Lehrer in [12]. It turns out that End KG(M )_ K[W], the Hecke algebra of W, where W = WJ denotes the Weyl group of CG(s) (the centralizer of s in G) (compare [1, 2, 4, 13]). In particular, K[W] has a basis { Tw, Iw c W } such that the structure constants of the multiplication are all contained in R c K. So let R[W] be the R-order in K[W] generated by wTIw EW} and R[W] = R ?R R[W]. Choosing a special RP-lattice S in M, and using the classification of the r-blocks of RG given by P. Fong and B. Srinivasan in Received by the editors July 25, 1983 and, in revised form, October 18, 1983, May 23, 1984 and September 1, 1984. 1980 Mathematics Subject Classification. Primary 20C20; Secondary 20G40. 'This work was supported by the Deutsche Forschungsgemeinschaft. ?1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page

Normal subgroup of the full linear group of degree 3 over a distributive lattice

Normal subgroup of the full linear group of degree 3 over a distributive lattice

MODELS FOR THE COMPLEX REPRESENTATIONS OF THE GROUPS $ \mathrm{GL}(n,\,q)$

The main result of the paper consists in the construction of a model of the full linear group over a finite field, i.e. its representations such that each irreducible representation occurs as a component precisely once. The series of representations thus constructed has the well-known Gel'fand-Graev representation as first term.Bibliography: 12 titles.

Finite automorphism groups of laminated near-rings

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

Some subgroups of the full linear group

Further observations are made on the author's earlier paper (Ref. Zh. Mat., 1977, 5A284) dealing with the lattice H of all subgroups of the full linear group GL(n, K) over a field K that contain the group K of diagonal matrices. It is noted, for example, that for an infinite field K all subgroups inD(n, K) are algebraic; a subgroup in H is connected if and only if it is a net subgroup; the lattice of all connected subgroups in H is isomorphic to the lattice of all marked topologies onn points; any subgroup H in H is a semidirect product H=A·Ho of a maximal connected normal subgroup Ho of H and a finite group A of, permutation matrices.

Parabolic subgroups of the full linear group over a Dedekind ring of arithmetical type

Suppose K is a global field, S a finite set of valuations of K containing all Archimedean valuations, and R the ring of S-integral elements of K. Assume that card S⩾2, R is generated by its invertible elements, and the ideal of R generated by the differences ε−1 for all invertible ɛ coincides with R. Under these assumptions, the parabolic subgroups of GL(n, R) are described. Namely, for each parabolic subgroup P there exists a unique net б of ideals of R (Ref. Zh. Mat., 1977, 2A280) such that e(б)⩽P⩽G(б), where G is the net subgroup of (б) and E(б) is the subgroup generated by the transvections in G(б). It is shown that E(б) is a normal subgroup of G(б). The factor group G(б/E(б)) is studied. The case of the special linear group is also considered.

Subgroups of the full linear group over the field of quotients of a Dedekind ring

We give a description of the lattice of all subgroups of the full linear group over the field of quotients of a Dedekind ring that contain the special linear group over this ring.

Subgroups of the full linear group over a Dedekind ring

We study the subgroups of the full linear group GL(n, R) over a Dedekind ring R that contain the group of quasidiagonal matrices of fixed type with diagonal blocks of at least third order, each of which is generated by elementary matrices. For any such subgroup H there exists a unique D-net σ of ideals of R such that , where E(σ) is the subgroup generated by all transvections of the net subgroup G(σ). and is the normalizer of G(σ). The subgroup E(σ) is normal in . To study the factor group we introduce an intermediate subgroup F(σ), E(σ) ≤ F(σ) ≤ G(σ). The group is finite and is connected with permutations in the symmetric group. The factor group G(σ)/F(σ) is Abelian — these are the values of a certain “determinant.” In the calculation of F(σ)/E(σ) appears the SK1-functor. Results are stated without proof.

Normal structure of the full linear group over a semilocal ring

It is shown how to obtain the known description of the normal subgroups of the full linear group over a semilocal ring from the results in Ref. Zh. Mat. 1979, 1A261.

Conjugacy of net subgroups of linear groups

For the full linear group over a matrix-local ring whose quotient by the Jacobson radical is not the field of two elements, we settle the question of the conjugacy ofD -net subgroups (Ref. Zh. Mat., 1977, 2A280). TwoD -net subgroups are conjugate if and only if theD -nets defining them are similar (i.e., can be transformed into each other by a permutation matrix). An analogous result is obtained forD -net subgroups of the symplectic group over a commutative local ring whose residue field contains more than three elements.

Self-normalizing nilpotent subgroups of the full linear group over a finite field

It has been proved (Ref. Zh. Mat., 1977, 4A170) that in the full linear group GL(n,q), n=2, 3, over a finite field of q elements, q odd or q=2, the only self-normalizing nilpotent subgroups are the normalizers of Sylow 2-subgroups and that for even q>2 there are no such subgroups. In the present note it is deduced from results of D. A. Suprunenko and R. F. Apatenok (Re. Zh. Mat., 1960, 13586; 1962, 9A150) that this is true for any n.

Description of subgroups of the full linear group over a semilocal ring that contain the group of diagonal matrices

It has been proved (Ref. Zh. Mat., 1978, 9A237) that for a semilocal ring Λ in which each residue field of the center contains at least seven elements we have the following description of subgroups of the full linear group GL(n,Λ) that contain the group of diagonal matrices: for each such subgroup H there is a uniquely definedD -net of idealsS (Ref. Zh. Mat., 1977, 2A288) such that G(S)≤H≤N(S), ,whereN(S) is the normalizer of theS -net subgroup G(S). It is noted that this result is also true under the following weaker assumption: a decomposition of a quotient ring of the ring Λ into a direct sum of full matrix rings over skew fields does not contain skew fields with centers of less than seven elements or the ring of second-order matrices over the field of two elements.

Subgroups of the full linear group over the field of quotients of a principal ideal domain

The structure is described of all subgroups of the full linear group over the field of quotients of a principal ideal domain R which contain the special linear group over R.

Irreducible representations of the full linear group

We obtain some results on Young diagrams. Based on these results, we construct bases for Young symmetry classes of tensors. Using these bases, we obtain a complete reduction of the representation A ↦⊗ m A [ A∈GL( n, C ] and irreducible matrix representations of the full linear group.

Parabolic congruence subgroups in linear groups

Parabolic subgroups are described for the full and special linear groups over a commutative ring R which contain a principal congruence level a, where a is an ideal of R such that R/a is semilocal. It is assumed that R is generated additively by its invertible elements and that the ring identity can be expressed as a sum of two invertible elements.

Normal net subgroups of the full linear group

We prove that for n ≥ 3 a net subgroup of the full linear group G=GL(n, Λ) over an arbitrary associative ring Λ with unity (see [1]) is normal in G if and only if it is a principal congruence subgroup. We also study the case n=2, where the situation is, in general, more complicated.