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In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn.
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In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.
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In mathematics, Cayley's Ω process, introduced by Arthur Cayley (1846), is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action.
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In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables x and y that remains invariant under the special linear group acting on the variables x and y.
Nolan Russell Wallach is a mathematician known for work in the representation theory of reductive algebraic groups. He is the author of the 2-volume treatise Real Reductive Groups.
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus T is called a T-variety. In differential geometry, one considers an action of a real or complex torus on a manifold.
In algebra, the first and second fundamental theorems of invariant theory concern the generators and the relations of the ring of invariants in the ring of polynomial functions for classical groups. The theorems are among the most important results of invariant theory.
In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra ; in particular, it gives a way to parametrize irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.
References
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