Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Composition of transpositions and equality of ribbon Schur Q-functions. | Composition of transpositions and equality of ribbon Schur Q-functions Farzin Barekat Stephanie van Willigenburg Department of Mathematics Department of Mathematics University of British Columbia University of British Columbia Vancouver BC V6T 1Z2 Canada Vancouver BC V6T 1Z2 Canada farzin_barekat@ steph@ Submitted Apr 1 2009 Accepted Aug 24 2009 Published Aug 31 2009 Mathematics Subject Classification Primary 05A19 05E10 Secondary 05A17 05E05 Keywords compositions Eulerian posets ribbons Schur Q-functions tableaux Abstract We introduce a new operation on skew diagrams called composition of transpositions and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur Q-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon we conjecture necessary and sufficient conditions for equality of ribbon Schur Q-functions. Moreover we determine all relations between ribbon Schur Q-functions show they supply a Z-basis for skew Schur Q-functions assert their irreducibility and show that the non-commutative analogue of ribbon Schur Q-functions is the flag h-vector of Eulerian posets. Contents 1 Introduction 2 2 Diagrams 3 Operations on diagrams . 4 Preliminary properties of . 6 3 Skew Schur Q-functions 6 Symmetric functions and 0. 8 New bases and relations in Q . 9 Equivalence of relations. 12 4 Equality of ordinary skew Schur Q-functions 15 5 Ribbon Schur Q-functions 22 Equality of ribbon Schur Q-functions. 23 THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R110 1 1 Introduction In the algebra of symmetric functions there is interest in determining when two skew Schur functions are equal 4 7 11 12 17 . The equalities are described in terms of equivalence relations on skew diagrams. It is consequently natural to investigate whether new equivalence relations on skew diagrams arise when we restrict our attention to the subalgebra of skew Schur Q-functions. This is a .
