Sami H. Assaf
C. L. E. Moore Instructor
Massachusetts Institute of
Technology
Office: Building 4, Room 182 |
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A generalized major index statistic. Seminaire Lotharingien de Combinatoire 60 (2008), Art. B60c, 13 pp. (electronic).Inspired by the k-inversion statistic for LLT polynomials, we define the k-inversion number and k-descent set for words. Using these, we define a new statistic on words, called the k-major index, that interpolates between the major index and inversion number. We give a bijective proof that the k-major index is equidistributed with the major index, generalizing a classical result of Foata and rediscovering a result of Kadell. Inspired by recent work of Haglund and Stevens, we give a partial extension of these definitions and constructions to standard Young tableaux. Finally, we give an application to Macdonald polynomials made possible by connections with LLT polynomials.
&bull 12 pages &bull preprint &bull link to final version &bull
A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs. Discrete Mathematics and Theoretical Computer Science, to appear as part of the conference proceedings for FPSAC 2008.For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.
&bull 10 pages &bull preprint &bull slides from MSRI Workshop (01/2008) &bull
The Schur expansion of Macdonald polynomials.Building on Haglund's combinatorial formula for the transformed Macdonald polynomials, we provide a purely combinatorial proof of Macdonald positivity using dual equivalence graphs and give a combinatorial formula for the coefficients in the Schur expansion.
&bull 9 pages &bull preprint &bull slides from BIRS Workshop (09/2007) &bull
A combinatorial proof of LLT and Macdonald positivity.We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomials.
&bull 24 pages &bull preprint &bull
Dual equivalence graphs, ribbon tableaux and Macdonald polynomials. Ph.D. dissertation, University of California Berkeley, May 2007.We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize such constructions and prove that the generating functions of these graphs are Schur positive. We construct a graph on k-ribbon tableaux which we conjecture to be a dual equivalence graph, and we prove the conjecture for k at most 3. This implies the Schur positivity of the k-ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's monomial expansion for Macdonald polynomials, this has the further consequence of a combinatorial Schur expansion of the transformed Macdonald polynomials indexed by a partition with at most 3 columns.
&bull 60 pages &bull PDF &bull 3-page synopsis &bull 9-page extended abstract &bull
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Workshop on
Representation Theory, Geometry and Combinatorics University of California Berkeley, Berkeley, CA June 2 - 6, 2008 FPSAC 2008, 20th International conference on Formal Power Series and Algebraic Combinatorics Valparaiso, Chile June 23 - 27, 2008 AMS Western Section Meeting University of British Columbia, Vancouver, Canada October 4 - 5, 2008 |
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| If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one. |
| --John Barrow |
