Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: An inverse matrix formula in the right-quantum algebra. | An inverse matrix formula in the right-quantum algebra Matjaz Konvalinka Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139 USA konvalinka@ http konvalinka Submitted Sep 20 2007 Accepted Jan 23 2008 Published Feb 4 2008 Mathematics Subject Classification 15A09 primary 05A15 secondary Abstract The right-quantum algebra was introduced recently by Garoufalidis Le and Zeilberger in their quantum generalization of the MacMahon master theorem. A bijective proof of this identity due to Konvalinka and Pak and also the recent proof of the right-quantum Sylvester s determinant identity make heavy use of a bijection related to the Erst fundamental transformation on words introduced by Foata. This paper makes explicit the connection between this transformation and right-quantum linear algebra identities we give a new bijective proof of the right-quantum matrix inverse theorem we show that similar techniques prove the right-quantum Jacobi ratio theorem and we use the matrix inverse formula to End a generalization of the right-quantum MacMahon master theorem. 1 Introduction Combinatorial linear algebra is a beautiful and underdeveloped part of enumerative combinatorics. The underlying idea is very simple one takes a matrix identity and views it as an algebraic result over a possibly non-commutative ring. Once the identity is translated into the language of words an explicit bijection or an involution is employed to prove the result. The resulting combinatorial proofs are often insightful and lead to extensions and generalizations of the original identities often in unexpected directions. A tremendous body of literature exists on quantum linear algebra . on quantum matrices. Without going into definitions history and technical details let us mention Manin s works Man88 Man89 . Recently the work of Garoufalidis Lê and Zeilberger GLZ06 suggested that certain linear algebra identities such as the celebrated MacMahon .
