Abstract
In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we use Panyushev's map, together with the well-known Kreweras complement, to construct a bijection between nonnesting and noncrossing partitions. Our map is defined uniformly for all root systems, using a recursion in which the map is assumed to be defined already for all parabolic subsystems. Unfortunately, the proof that our map is well defined, and is a bijection, is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectural properties of the Panyushev map, and we prove two cyclic sieving phenomena conjectured by D. Bessis and V. Reiner. 2013 American Mathematical Society.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 4121-4151 |
| Number of pages | 31 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 365 |
| Issue number | 8 |
| DOIs | |
| State | Published - 2013 |
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Keywords
- Bijective combinatorics
- Coxeter groups
- Cyclic sieving phenomenon
- Noncrossing partitions
- Nonnesting partitions
- Weyl groups
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
A uniform bijection between nonnesting and noncrossing partitions. / Armstrong, Drew; Stump, Christian; Thomas, Hugh.
In: Transactions of the American Mathematical Society, Vol. 365, No. 8, 2013, p. 4121-4151.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - A uniform bijection between nonnesting and noncrossing partitions
AU - Armstrong, Drew
AU - Stump, Christian
AU - Thomas, Hugh
PY - 2013
Y1 - 2013
N2 - In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we use Panyushev's map, together with the well-known Kreweras complement, to construct a bijection between nonnesting and noncrossing partitions. Our map is defined uniformly for all root systems, using a recursion in which the map is assumed to be defined already for all parabolic subsystems. Unfortunately, the proof that our map is well defined, and is a bijection, is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectural properties of the Panyushev map, and we prove two cyclic sieving phenomena conjectured by D. Bessis and V. Reiner. 2013 American Mathematical Society.
AB - In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we use Panyushev's map, together with the well-known Kreweras complement, to construct a bijection between nonnesting and noncrossing partitions. Our map is defined uniformly for all root systems, using a recursion in which the map is assumed to be defined already for all parabolic subsystems. Unfortunately, the proof that our map is well defined, and is a bijection, is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectural properties of the Panyushev map, and we prove two cyclic sieving phenomena conjectured by D. Bessis and V. Reiner. 2013 American Mathematical Society.
KW - Bijective combinatorics
KW - Coxeter groups
KW - Cyclic sieving phenomenon
KW - Noncrossing partitions
KW - Nonnesting partitions
KW - Weyl groups
UR - http://www.scopus.com/inward/record.url?scp=84877974174&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84877974174&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-2013-05729-7
DO - 10.1090/S0002-9947-2013-05729-7
M3 - Article
AN - SCOPUS:84877974174
VL - 365
SP - 4121
EP - 4151
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 8
ER -