Mathematical Excursion. Group Characters

Abstract

In this chapter a special group theoretical concept is introduced which has many applications. It describes the main properties of representations and is therefore called “group character”. It solves the problem of how to describe the invariant properties of a group Ĝa , representation in a simple way. If we denote an element of a group G by Ĝ _a representation Ď(Ĝ _a) is not unambiguous, because every similarity transformation ÂĎ(Ĝ_a)Â^-1, Â ∈ D(G) yields an equivalent form. One possibility for the description of the invariant properties would be to use the eigenvalues of the representation matrix, which do not change under a similarity transformation. This leads to the construction of the Casimir operators, the eigenvalues of which classify the representations. The construction of the Casimir operators and their eigenvalues is in general a very difficult nonlinear problem. Fortunately, in many cases it is sufficient to use a simpler invariant, namely the trace of the representation matrix

EquationSource%% MathType!Translator!2!1!LaTeX.tdl!TeX -- LaTeX 2.09 and later! % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaacI % caceWGhbGbambadaWgaaWcbaGaamyyaaqabaGccaGGPaGaeyypa0Za % aabCaeaacaWGebWaaSbaaSqaaiaadMgacaWGPbaabeaakiaacIcace % WGhbGbambadaWgaaWcbaGaamyyaaqabaGccaGGPaaaleaacaWGPbGa % eyypa0JaaGymaaqaaiaadsgaa0GaeyyeIuoaaaa!4740! \[X({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over G} _a}) = \sum\limits_{i = 1}^d {{D_{ii}}({{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over G} }_a})} \] $$

((10.1))

where d is the dimension of the matrix representation. Equation (10.1) is in fact invariant under similarity transformations, because

EquationSource%% MathType!Translator!2!1!LaTeX.tdl!TeX -- LaTeX 2.09 and later! % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbau % aacaGGOaGabm4rayaataWaaSbaaSqaaiaadggaaeqaaOGaaiykaiab % g2da9maaqafabaGabmirayaafaWaaSbaaSqaaiaadMgacaWGPbaabe % aaaeaacaWGPbaabeqdcqGHris5aOGaaiikaiqadEeagaWeamaaBaaa % leaacaWGHbaabeaakiaacMcacqGH9aqpdaaeqbqaaiaadgeadaWgaa % WcbaGaamyAaiaadQgaaeqaaOGaamiramaaBaaaleaacaWGQbGaam4A % aaqabaGccaGGOaGabm4rayaataWaaSbaaSqaaiaadggaaeqaaOGaai % ykaiaacIcaceWGbbGbambadaahaaWcbeqaaiabgkHiTiaaigdaaaGc % caGGPaaaleaacaWGPbGaamOAaiaadUgaaeqaniabggHiLdGccaWGRb % GaamyAaiabg2da9maaqafabaGaamiramaaBaaaleaacaWGQbGaamOA % aaqabaGccaGGOaGabm4rayaataWaaSbaaSqaaiaadggaaeqaaOGaai % ykaaWcbaGaamOAaiaadUgaaeqaniabggHiLdGccqGH9aqpcaWGybGa % aiikaiqadEeagaWeamaaBaaaleaacaWGHbaabeaakiaacMcaaaa!6B05! \[\psi '({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over G} _a}) = \sum\limits_i {{{D'}_{ii}}} ({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over G} _a}) = \sum\limits_{ijk} {{A_{ij}}{D_{jk}}({{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over G} }_a})({{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over A} }^{ - 1}})} ki = \sum\limits_{jk} {{D_{jj}}({{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over G} }_a})} = X({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over G} _a})\] $$

((10.2))

X(Ĝ_a) is called the “grup character” of the representation.