CONTRIBUTIONS TO THE HISTORY OF DETERMINANTS

By X. RANIM

CONTENTS MULTIPLICATIVE FORMS ON ALGEBRAS AND THE GROUP DETERMINANT INTRODUCTION SUMS OF SQUARES AND NORM FORMS ON ALGEBRAS DIOPHANTINE EQUATIONS AND THE COMPOSITION OF QUADRATIC FORMS THE RELATION TO THE THEORY OF ALGEBRAS GROUP MATRICES AND THE GROUP DETERMINANT GROUP CHARACTERS BEFORE FROBENIUS THE RESULTS ON CHARACTERS WHICH APPEAR IN [106] AND [107] GROUP DETERMINANT FACTORS GROUP MATRICES RIGHT DIVISION IN GROUPS AND GROUP MATRICES OF SMALL GROUPS SYMMETRICAL FORMS OF THE GROUP MATRIX THE GROUP DETERMINANT THE ORIGINAL MOTIVATION FOR THE EXAMINATION OF THE GROUP DETERMINANT FROBENIUS’ DE?NITION OF GROUP CHARACTERS GROUP CHARACTERS AND THE GROUP DETERMINANT FROBENIUS’ BASIC RESULTS AND THEIR PROOFS SCHUR’S APPROACH TO REPRESENTATION THEORY K-CHARACTERS AND THE CONSTRUCTION OF IRREDUCIBLE FACTORS THE CONNECTION WITH THE CAYLEY-HAMILTON THEOREM AND SYMMETRIC FUNCTIONS HISTORICAL REMARKS EARLY GROUP DETERMINANT WORK AFTER FROBENIUS THE INITIAL RESULTS ON THE INFORMATION IN K-CHARACTERS SUMMARY OF FURTHER DEVELOPMENTS FURTHER GROUP MATRICES AND GROUP DETERMINANTS ARBITRARY GROUP MATRICES A CONNECTION WITH INVARIANT THEORY REPRESENTATION OF GROUP MATRICES IN BLOCK FORM GROUP MATRICES BUILT FROM ARBITRARY SUBGROUPS GROUP MATRICES OF DIRECT PRODUCTS GROUPS WITH A NORMAL SUBGROUP SPLIT EXTENSIONS GROUPS WHICH HAVE AN EXACT FACTORIZATION AS A PRODUCT OF TWO SUBGROUPS THE GROUP MATRIX USING CONJUGACY CLASS ORDERING A DIGRESSION: THE OCTONION LOOP METHODS TO FACTORIZE THE GROUP DETERMINANT EXPLICIT DIAGONALIZATION OF A GROUP MATRIX THE PARTIAL DIAGONALIZATION WITH RESPECT TO A GIVEN SUBGROUP INDUCTION AND RESTRICTION OF REPRESENTATIONS METHODS FROM GENERAL DETERMINANT THEORY THE SCHUR COMPLEMENT MORE EXAMPLES VIRTUAL REPRESENTATIONS VIA SUPERMATRICES SUPERALGEBRAS AND SUPERMATRICES GROUP MATRICES CORRESPONDING TO VIRTUAL REPRESENTATIONS THE HOPF ALGEBRA OPERATIONS GROUP MATRICES AND THE BURNSIDE RING PROJECTIVE REPRESENTATIONS AND GROUP MATRICES THE CONNECTION WITH THE GROUP DETERMINANT GROUP MATRICES AND GROUP DETERMINANTS IN THE MODULAR CASE DICKSON’S CONSTRUCTION OF THE GROUP MATRIX AS A BLOCK MATRIX NORM FORMS AND GROUP DETERMINANT FACTORS NORM-TYPE FORMS ON ALGEBRAS MULTIPLICATIVE PROPERTIES OF NORM FORMS ON ALGEBRAS MULTILINEARIZATION RESULTS ON MULTIPLICATIVE NORM FORMS ON ALGEBRAS THE CUBIC FORM AND THE JORDAN ALGEBRA ARBITRARY MULTIPLICATIVE FORMS ON ALGEBRAS THE RELATIONSHIP TO THE WORK OF THE SCHOOL OF A BERGMANN THE CONNECTION WITH THE “AVERAGE” INVARIANTS OF ROITMAN S-RINGS, GELFAND PAIRS AND ASSOCIATION SCHEMES CENTRALIZER RINGS OF PERMUTATION GROUPS S-RINGS PERMUTATION GROUPS WITH LOOP TRANSVERSALS COMMUTATIVITY OF THE CENTRALIZER RING HECKE ALGEBRAS ASSOCIATION SCHEMES CHARACTER TABLES OF ASSOCIATION SCHEMES DUALITY IN ASSOCIATION SCHEMES FUSION OF CHARACTER TABLES THE MAGIC RECTANGLE CONDITION INDUCTION AND FROBENIUS RECIPROCITY FOR ASSOCIATION SCHEMES SOME OTHER APPEARANCES OF ASSOCIATION SCHEMES QUASIGROUP CHARACTERS THE MULTIPLICATION GROUP OF A QUASIGROUP CLASS FUNCTIONS ON QUASIGROUPS THE INFORMATION IN THE CHARACTER TABLE OF A LOOP OR QUASIGROUP THE INFORMATION IN THE CHARACTER TABLE OF A LOOP THE INFORMATION IN THE CHARACTER TABLE OF A QUASIGROUP RANK 2 QUASIGROUPS INDUCTION AND RESTRICTION FOR QUASIGROUP CHARACTERS AND FROBENUS RECIPROCITY A GENERALIZATION OF ARTIN’S THEOREM FOR GROUP CHARACTERS METHODS OF CALCULATION THE 2-CHARACTERS OF A GROUP AND THE WEAK CAYLEY TABLE THE INFORMATION IN THE 1- AND 2-CHARACTERS OF A GROUP THE INFORMATION CONTAINED IN THE ORDINARY CHARACTER TABLE THE WEAK CAYLEY TABLE OF A GROUP THE 2-CHARACTER TABLE PROPERTIES DETERMINED BY THE WEAK CAYLEY TABLE GROUPS WITH THE SAME WCT WEAK CAYLEY ISOMORPHISMS AND WEAK CAYLEY TABLE GROUPS FINITE GROUPS WITH TRIVIAL WEAK CAYLEY GROUPS GROUPS WITH NON-TRIVIAL W(G) SIMPLE FINITE GROUPS IN?NITE GROUPS THE CATEGORY OF WEAK CAYLEY MORPHISMS THE CATEGORY OF WEAK MORPHISMS THE ADJUNCTION HOMOTOPY THE EXTENDED K-CHARACTERS K-CLASSES AND EXTENDED K-CHARACTERS K-CLASSES RELEVANT REPRESENTATION THEORY OF SN OVER C THE CONNECTION WITH WREATH PRODUCTS ?? AS A MULTILINEARIZATION ?? AS A TRACE AND AN INVARIANT ?? AS AN INVARIANT K-CLASS FUNCTIONS ORTHOGONALITY THE NUMBER OF K-CLASSES AND EXTENDED K-FUNCTIONS FOURIER ANALYSIS ON GROUPS, RANDOM WALKS AND MARKOV CHAINS FOURIER ANALYSIS ON GROUPS THE FOURIER TRANSFORM FROM FACTORS OF THE GROUP DETERMINANT INDUCED CLASS FUNCTIONS AND CONDITIONAL EXPECTATIONS PROBABILITY ON ASSOCIATION SCHEMES FINITE FOURIER TRANSFORMS THE DISCRETE FOURIER TRANSFORM ON ZN THE FAST FOURIER TRANSFORM: AN EXPLANATION IN TERMS OF THE GROUP MATRIX FAST FOURIER TRANSFORMS FOR ARBITRARY FINITE GROUPS MARKOV CHAINS AND RANDOM WALKS THE BASIC DE?NITION RANDOM WALKS ON GROUPS RANDOM WALKS THAT BECOME UNIFORM IN A FINITE NUMBER OF STEPS ARBITRARY RANDOM WALKS ON GROUPS FISSION OF GROUP CLASSES HOMOGENEOUS ACTIONS OF FINITE GROUPS AN EXAMPLE K-CHARACTERS AND N-HOMOMORPHISMS FROBENIUS N-HOMOMORPHISMS AND GEOMETRY FROBENIUS N-HOMOMORPHISMS AND RELATED OBJECTS A GENERALIZATION OF THE GELFAND–KOLMOGOROV THEOREM THE FKG INEQUALITY AND ITS GENERALIZATION EXTENSIONS TO HIGHER CUMULANTS THE CONJECTURE OF SAHI THE CONNECTION WITH K-CHARACTERS FUSION AND SUPERCHARACTERS FUSION FROM ABELIAN GROUPS BASIC RESULTS ON FUSION RESULTS ON GROUPS WHICH LIE IN FA RESULTS ON GROUPS WHICH ARE NOT IN FA GROUPS WHICH FUSE FROM A CYCLIC GROUP SUPERCHARACTERS OF UPPER TRIANGULAR GROUPS SYMMETRIC FUNCTIONS IN NON-COMMUTING VARIABLES A FUSION USING NONASSOCIATIVE ALGEBRA OTHER SITUATIONS INVOLVING GROUP MATRICES CHARACTERIZATION OF GROUP CHARACTERS THE FORMAL POWER SERIES ATTACHED TO A CLASS FUNCTION PSEUDOREPRESENTATIONS GROUP MATRICES AND GROUP RINGS TAUSSKY-TODD ON NORMAL GROUP MATRICES OTHER EARLY WORK ON GROUP MATRICES THE INTEGRAL GROUP RING CASE SOME RINGS OF MATRICES WHICH OCCUR AS GROUP RINGS GROUP MATRICES AND COGROWTH OF GROUPS COGROWTH OF FINITE ABELIAN GROUPS THE CONNECTION BETWEEN COGROWTH AND THE GROUP MATRIX POINCARÉ AND THE GROUP DETERMINANT CHERN CLASSES TIGHT FRAMES AND WAVELETS FRAMES AND GRAMIAN MATRICES GROUP FRAMES PROJECTIVE GROUP MATRICES AND FINITE FRAMES THREE-MANIFOLDS GROUP MATRICES AND ELECTRICAL ENGINEERING HIGHER S-RINGS