Projects / Programmes
Pieri rules for Hall-Littlewood polynomials and Macdonald polynomials
| Code |
Science |
Field |
Subfield |
| 1.01.05 |
Natural sciences and mathematics |
Mathematics |
Graph theory |
| Code |
Science |
Field |
| P110 |
Natural sciences and mathematics |
Mathematical logic, set theory, combinatories |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
Pieri rules, symmetric functions, Schur functions, Hall-Littlewood polynomials, Macdonald polynomials, k-Schur functions
Researchers (1)
| no. |
Code |
Name and surname |
Research area |
Role |
Period |
No. of publicationsNo. of publications |
| 1. |
22401 |
PhD Matjaž Konvalinka |
Mathematics |
Head |
2013 - 2015 |
118 |
Organisations (1)
Abstract
Schur functions form the most important basis of the algebra of symmetric functions. They are important not just because they are orthonormal, but also because of the connections to the representation theory of finite groups and other areas of combinatorics and algebra. Their history started in the 19th century and is tightly intertwined with the development of many branches of mathematics. The study of their properties and generalizations continues today.
A key result is the Littlewood-Richardson rule, which expresses the product of Schur functions in terms of Schur functions. The rule is complicated and was not proved until the 70s. The Pieri rule is a special case and is much simpler, see (7) in the attached file.
Pieri rule has analogues for Hall-Littlewood polynomials (see (9)), Macdonald P-polynomials (see (10)) and k-Schur functions (see (11)).
The Pieri rule has many generalizations, but one area of research remained unexplored until recently. Assaf and McNamara [1] found a way to expand the product of a skew Schur function with sr and s1...1 in terms of skew Schur functions. I generalized their elegant result in [2]; the result expands the product of a Schur function with the Hall-Littlewood polynomial Pr(t). See (13).
I would like to answer the following questions.
1. In [2], I stated two conjectures about the product of a skew Hall-Littlewood polynomial with Schur functions sr and s1...1, see (14), (15). I would like to prove them. (14) gives a new result even in the non-skew case. I assume that the results can be proved via the Hopf algebra method due to Lam, Lauve and Sotille (2010). I would like to find an involutive proof as well.
2. Computer computations show that the polynomials that appear in the product of a skew Hall-Littlewood polynomial with Pr(t) are much more complicated than the ones in the product with sr or s1...1. More precisely, it seems that in the factorization in Z[t], the irreducible factors can be of arbitrarily high degree, but with small integer coefficients. I would like to study the properties of these polynomials and find a formula.
3. As stated above, there is a variant of the Pieri rule for Macdonald P-polynomials, where the coefficients are rational functions. Because of that, I have not been able to state conjectures about the product of a skew Macdonald polynomial with any of the important functions (e.g. sr, Pr(t), Pr(q,t)). It would also be interesting to find a variant of the (skew) Pieri rule for Macdonald H-polynomials, which can be expressed in terms of Schur functions with coefficients that are polynomials in q and t.
4. Pieri rule for k-Schur functions has a natural skew analogue. The main tool in [1] and [2] is insertion. A version for cores was found by Lam, Lapointe, Morse and Shimozono. I would like to find a variant for bounded partitions and use it to find an involutive proof.
5. I would like to try to find the Littlewood-Richardson rule for k-Schur functions, at least in some special cases.
The answers will be of interest to the combinatorial community and other mathematicians. I plan to present my work in high-quality journals and at the most important annual conference in the fields of algebraic and enumerative combinatorics, Formal Power Series and Algebraic Combinatorics. I have presented my work there every year since 2007. I plan to collaborate with mathematicians abroad; the project would give me the necessary time to visit them.
[1] S. H. Assaf, P. R. W. McNamara, A Pieri rule for skew shapes, J. Combin. Theory Ser. A, Vol. 118 No. 1 (2011), 277-290
[2] M. Konvalinka, Skew quantum Murnaghan-Nakayama rule, J. Algebraic Combin., Vol. 35 (4) (2012), 519-545.
[3] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Second edition, The Clarendon Press Oxford University Press, New York, 1995
Significance for science
In the project, I studied residue and quotient tables, and they have the potential to solve many open problems in the area of k-Schur functions and related area (e.g. Gromov-Witten invariants). Tanglegrams are an important tool in biology and theoretical computer science. Our results and methods have the potential to solve many problems in the area, and have been an inspiration for at least 6 papers. Alternating sign matrices became a mathematical area in the eighties. One of the first papers, by D. Robbins, introduced several conjectures about the cardinality of special subclasses of these objects. The methods developed in solving these problems have proved instrumental in the area. In the paper with Roger Behrend and Ilse Fischer, we proved the last of these conjectures and thus closed an important chapter of an important mathematical field.
Significance for the country
The project has given me, my university and Slovenia more recognition. Based on my research, several papers were published in international journals, and I have presented and will present my results at international conferences. For example, I will be a coauthor of there papers at the conference FPSAC 2016 in Vancouver, the most prestigious annual conference in my area. I was also a coadvisor for a Ph.D. thesis and an advisor for two Master's thesis, thereby contributing to the development of young Slovenian mathematicians.
Most important scientific results
Annual report
2013,
2014,
final report,
complete report on dLib.si
Most important socioeconomically and culturally relevant results
Annual report
2014,
final report,
complete report on dLib.si
