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 ??^mA [A∈GL(n,$\text{C}$] and irreducible matrix representations of the full linear group.     

Irreducibility criterion for regular Gaussian representations of groups of finite upper triangular matrices

Institute of Mathematics, Academy of the Sciences of the USSR. Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 24, No. 3, pp. 82–83, July–September, 1990.     

The group of commutativity preserving maps on strictly upper triangular matrices

Let N = N _n (R) be the algebra of all n × n strictly upper triangular matrices over a unital commutative ring R. A map φ on N is called preserving commutativity in both directions if xy = yx ? φ(x)φ(y) = φ(y)φ(x). In this paper, we prove that each invertible linear map on N preserving commutativity in both directions is exactly a quasi-automorphism of N, and a quasi-automorphism of N can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.     

Derivations of the Lie algebra of infinite strictly upper triangular matrices over a commutative ring

Let 𝒩(∞,R) be the Lie algebra of infinite strictly upper triangular matrices over a commutative ring R. We show that every derivation of 𝒩(∞,R) is a sum of diagonal and inner derivations.     

Finite traces and representations of the group of infinite matrices over a finite field

The article is devoted to the representation theory of locally compact infinite-dimensional group GLB$\mathbb{GLB}$ of almost upper-triangular infinite matrices over the finite field with q   elements. This group was defined by S.K., A.V., and Andrei Zelevinsky in 1982 as an adequate n=∞$n=\infty$ analogue of general linear groups GL(n,q)$\mathbb{GL}(n,q)$. It serves as an alternative to GL(∞,q)$\mathbb{GL}(\infty ,q)$, whose representation theory is poor.     

Three linear preservers on infinite triangular matrices

We consider three linear preserver problems on the algebra of infinite triangular matrices over fields. We characterize the maps preserving invertible and noninvertible matrices, the surjective maps preserving inverses and the surjective maps preserving rank permutability.     

Spectra of upper triangular operator matrices

Let be given Banach spaces. For and , let be the operator defined on by . We give sufficient conditions on to get where runs over a large class of spectra. We also discuss the case of some spectra for which the latter equality fails.

     

Factor representations of the infinite spin-symmetric group

Let S() be the group of finite permutations of the sequence of natural numbers. The infinite spin-symmetric group T() is its central Z₂-extension. With the aid of this extension the projective representations of the group S() can be linearized. The paper describes the factor representation of type II₁ of the group T().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 132–145, 1990.     

Irreducible representations of the alternating group in odd characteristic

We use the recently-proved conjecture of Mullineux to determine which modular irreducible representations of the symmetric group split on restriction to , and which remain irreducible (everything taking place over a splitting field for of characteristic ). An indexing of the absolutely irreducible representations of is thus obtained. A modular analogue of the Frobenius symbol for a partition is introduced, which makes the Mullineux map somewhat more intuitive.

     

Group gradings on upper triangular matrices

Let UT_n be the algebra of n × n upper triangular matrices over a field F. We describe all G-gradings on UT _n by an arbitrary group G. Received: 13 September 2006     

Preserving diagonalisability on upper triangular matrices

We characterise all bijective linear mappings on the algebra of upper triangular _n × n matrices that preserve diagonalisability.