# Workshop on Cylindric Partitions

## November 21 – 25, 2022

This is a research in teams event that will take place at Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences (OEAW) and Research Institute for Symbolic Computation (RISC) of Johannes Kepler University (JKU). Some talks are announced below to introduce the problems and the state of the art.

This event is supported by Doktoral Program Computational Mathematics at JKU.

## Confirmed Participants

• Zafeirakis Zafeirakopoulos, 21 Nov 2022 – 2pm @RICAM Seminar Room (SP2 416-2)
Polyhedral Omega (+ Applications)
Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decomposition and Barvinok’s short rational function representations. This synthesis of ideas makes Polyhedral Omega by far the simplest algorithm for solving linear Diophantine systems available to date. After presenting the algorithm, we will see some applications and generalizations.
• Christian Krattenthaler, 22 Nov 2022 – 2pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
Identities for cylindric Schur functions
A well-known, but difficult-to-prove result is the determinantal formula for the sum of Schur functions over shapes with a restricted number of columns, due to Gordon, as realised by Bender and Knuth. I will represent affine refinements of these identities, which can be seen as identities for cylindric Schur functions.
This is joint work with JiSum Huh, Jang Soo Kim and Soichi Okada.
• Ole Warnaar, 23 Nov 2022 – 2pm @ RISC Seminar Room
Cylindric partitions
Cylindric partitions are an affine analogue of plane partitions. They were first introduced in 1997 by Gessel and Krattenthaler, and are closely related to the representation theory of the affine Lie algebra $\mathrm{A}_{r-1}^{(1)}$. In this talk I will try to explain why cylindric partitions have become such a powerful combinatorial tool for discovering new identities of the Rogers–Ramanujan type.
• Jehanne Dousse, 24 Nov 2022 – 1:30pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
Cylindric partitions and mod 8 Rogers-Ramanujan type identities
Cylindric partitions, which were introduced by Gessel and Krattenthaler in 1997, can be seen as a generalisations of integer partitions involving periodicity conditions. Since the 1980s and the founding work of Lepowsky and Wilson on Rogers-Ramanujan identities, several connections between representation theory and partition identities have emerged. In particular, Andrews, Schilling and Warnaar discovered in 1998 a family of partition identities related to characters of A_2. Recently, Corteel and Welsh established a q-difference equation satisfied by generating functions for cylindric partitions, and used it to reprove the A_2 Rogers-Ramanujan identities of Andrews, Schilling and Warnaar. In this talk, we we build on this technique to discover and prove a new family of A_2 Rogers-Ramanujan identities with modulo 8 congruence conditions.
This is joint work with Sylvie Corteel and Ali Uncu.
• Shunsuke Tsuchioka, 24 Nov 2022 – 2:45pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
A Fibonacci variant of Rogers-Ramanujan identities via crystal energy
We define a length function for a perfect crystal. As an application, we derive a variant of the Rogers-Ramanujan identities which involves (a q-analog of) the Fibonacci numbers.
• Shashank Kanade, 24 Nov 2022 – 4pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
On the Andrews-Schilling-Warnaar identities
Generalizing the usual $A_1$ Bailey machinery, Andrews-Schilling-Warnaar discovered an $A_2$ generalization of the notion of Bailey pairs and the Bailey lemma. Using this, they discovered identities involving principal characters of standard $\widehat{\mathfrak{sl_3}}$ modules. However, for every positive integral level l (barring a few low-lying cases), a majority of identities involving level l standard $\widehat{\mathfrak{sl_3}}$ modules were yet to be discovered. In a recent joint work with Russell, we have given conjectures that encompass all of the missing identities. Especially, we can prove our conjectures in levels 3 and 7 (and also levels 2, 4, 5, but these cases are not exactly new). Our proofs crucially use the Corteel-Welsh recursions governing cylindric partitions. Using a different circle of ideas, we were also able to produce various results when the levels are divisible by 3. In this talk, I will explain these developments. Time permitting, I will touch upon some exciting open problems that still remain.
• Walter Bridges, 25 Nov 2022 – 10am @ RICAM Seminar Room (SP2 416-2) / Over Zoom
Weighted cylindric partitions and sum-product identities
Corteel and Welsh recently showed how sum-product identities arise naturally from cylindric partitions.  Using work of Han and Xiong, we extend their ideas to more general structures like weighted cylindric partitions, symmetric cylindric partitions and skew double-shifted plane partitions.  In this greater generality, “all products” appear.  We discover new identities and reprove the Göllnitz–Gordon and Little Göllnitz identities, as well as some so-called Schmidt-type partition identities highlighted in recent work of Andrews and Paule.
This is joint work with Ali Uncu.
• Ali Uncu, 25 Nov 2022 – 2pm @ RICAM Seminar Room (SP2 416-2) / Over Zoom
On modulo 11 and 13 cylindric partition conjectures of Kanade-Russell
In this concluding (and to be informal) talk, I will show how we recently proved Modulo 11 and 13 cylindrical partition related conjectures of Kanade-Russell.