The Properties And Classification Of Thin Inverting Pairs

Let F denote a field, V denote a vector space over F with finite positive dimension d+1and Matd+1(F) denote the F-algebra consisting of all d+1by d+1matrices that have entries in F. Assume A denotes an F-algebra isomorphic to Matd+1(F).Let V be an irreducible left A-module. By a thin inverting pair on V, We mean an ordered pair of invertible linear transformations on V, K:Vâ†'V, K*:Vâ†'V, that satisfies the following conditions:(â…°) There exists a basis for V with respect to which the matrix representing K has all entries0above the super-diagonal, has all entries nonzero on the super-diagonal, the matrix representing K-1has all entries0below the sub-diagonal, has all entries nonzero on the sub-diagonal, and the matrix representing K*is diagonal.(â…±) There exists a basis for V with respect to which the matrix representing K*has all entries0above the super-diagonal, has all entries nonzero on the super-diagonal, the matrix representing K*-1has all entries0below the sub-diagonal, has all entries nonzero on the sub-diagonal, and the matrix representing K is diagonal.A thin inverting pair on V is also called a thin inverting pair in A.This thesis mainly discuss the properties of the thin inverting pairs and give their classifi-cation. It is divided into the following five parts:In the first part, we introduce the notions of the thin inverting pair K, K*and the thin inverting system and discuss the corresponding relation between thin inverting pairs and thin inverting systems.In the second part, we show that V is irreducible as a (K, K*)-module. Moreover, we obtain the structure of A as a vector space through the thin inverting system in A.In the third part, for a thin inverting system Φ, we prove that there exists a split decompo-sition of VIn the fourth part, we define the parameter array of a thin inverting system Φ through the split decomposition of V, and prove that two thin inverting systems are isomorphic if and only if their parameter arrays are the same. Based on this we get the classification theorem of the thin inverting systems.In the fifth part, we give some formulas for (?)i and φi.