Conjectures about certain parabolic Kazhdan--Lusztig polynomials

Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan--Lusztig polynomials of type $A$. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan--Lusztig polynomials known as parabolic Kazhdan--Lusztig polynomials satisfy properties analogous to those of the ordinary ones.Comment: final versio

A decade of the berkeley math circle: the american experience, volume II

Many mathematicians have been drawn to mathematics through their experience with math circles. The Berkeley Math Circle (BMC) started in 1998 as one of the very first math circles in the U.S. Over the last decade and a half, 100 instructors--university professors, business tycoons, high school teachers, and more--have shared their passion for mathematics by delivering over 800 BMC sessions on the UC Berkeley campus every week during the school year. This second volume of the book series is based on a dozen of these sessions, encompassing a variety of enticing and stimulating mathematical topics, some new and some continuing from Volume I: from dismantling Rubik's Cube and randomly putting it back together to solving it with the power of group theory; from raising knot-eating machines and letting Alexander the Great cut the Gordian Knot to breaking through knot theory via the Jones polynomial; from entering a seemingly hopeless infinite raffle to becoming friendly with multiplicative functions in the land of Dirichlet, Möbius, and Euler; from leading an army of jumping fleas in an old problem from the International Mathematical Olympiads to improving our own essay-writing strategies; from searching for optimal paths on a hot summer day to questioning whether Archimedes was on his way to discovering trigonometry 2000 years ago Do some of these scenarios sound bizarre, having never before been associated with mathematics? Mathematicians love having fun while doing serious mathematics and that love is what this book intends to share with the reader. Whether at a beginner, an intermediate, or an advanced level, anyone can find a place here to be provoked to think deeply and to be inspired to create. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and theAMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession

Explicit enumeration of 321, hexagon-avoiding permutations

AbstractThe 321, hexagon-avoiding (321-hex) permutations were introduced and studied by Billey and Warrington (J. Alg. Comb. 13 (2001) 111–136). as a class of elements of Sn whose Kazhdan–Lusztig polynomials and the singular loci of whose Schubert varieties have certain fairly simple and explicit descriptions. This paper provides a 7-term linear recurrence relation leading to an explicit enumeration of the 321-hex permutations. A complete description of the corresponding generating tree is obtained as a by-product of enumeration techniques used in the paper, including Schensted's 321-subsequences decomposition, a 5-parameter generating function and the symmetries of the octagonal patterns avoided by the 321-hex permutations