Identifier
Images
['A',1] => ([],1) => ([],1) => ([],1)
['A',2] => ([(0,2),(1,2)],3) => ([(0,1),(0,2)],3) => ([],1)
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(0,3),(3,1),(3,2)],4) => ([],1)
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => ([],1)
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => ([],1)
['B',3] => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9) => ([(0,6),(3,8),(4,5),(4,8),(5,1),(5,7),(6,3),(6,4),(8,2),(8,7)],9) => ([],1)
['C',3] => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9) => ([(0,6),(3,8),(4,5),(4,8),(5,1),(5,7),(6,3),(6,4),(8,2),(8,7)],9) => ([],1)
['A',4] => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10) => ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10) => ([],1)
['B',4] => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16) => ([(0,9),(3,15),(4,14),(5,6),(5,14),(6,7),(6,10),(7,8),(7,13),(8,2),(8,11),(9,4),(9,5),(10,13),(10,15),(13,11),(13,12),(14,3),(14,10),(15,1),(15,12)],16) => ([],1)
['C',4] => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16) => ([(0,9),(3,15),(4,14),(5,6),(5,14),(6,7),(6,10),(7,8),(7,13),(8,2),(8,11),(9,4),(9,5),(10,13),(10,15),(13,11),(13,12),(14,3),(14,10),(15,1),(15,12)],16) => ([],1)
['D',4] => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12) => ([(0,7),(1,9),(1,10),(2,8),(2,10),(3,8),(3,9),(7,1),(7,2),(7,3),(8,6),(8,11),(9,4),(9,11),(10,5),(10,11)],12) => ([],1)
['F',4] => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24) => ([(0,12),(1,18),(1,19),(4,21),(5,15),(6,14),(7,13),(8,9),(8,21),(9,6),(9,22),(10,2),(10,16),(11,3),(11,17),(12,7),(13,4),(13,8),(14,18),(14,20),(15,19),(15,20),(18,10),(18,23),(19,11),(19,23),(20,23),(21,5),(21,22),(22,1),(22,14),(22,15),(23,16),(23,17)],24) => ([],1)
['A',5] => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15) => ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(12,14),(13,11),(13,14)],15) => ([],1)
['B',5] => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25) => ([(0,12),(2,13),(4,15),(5,16),(6,10),(6,13),(7,8),(7,18),(8,11),(8,19),(9,7),(9,17),(10,9),(10,20),(11,1),(11,14),(12,2),(12,6),(13,5),(13,20),(15,3),(15,22),(16,4),(16,24),(17,18),(17,24),(18,19),(18,21),(19,14),(19,23),(20,16),(20,17),(21,22),(21,23),(24,15),(24,21)],25) => ([],1)
['C',5] => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25) => ([(0,12),(2,13),(4,15),(5,16),(6,10),(6,13),(7,8),(7,18),(8,11),(8,19),(9,7),(9,17),(10,9),(10,20),(11,1),(11,14),(12,2),(12,6),(13,5),(13,20),(15,3),(15,22),(16,4),(16,24),(17,18),(17,24),(18,19),(18,21),(19,14),(19,23),(20,16),(20,17),(21,22),(21,23),(24,15),(24,21)],25) => ([],1)
['D',5] => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20) => ([(0,10),(1,15),(2,16),(2,17),(6,11),(6,13),(7,12),(7,13),(8,5),(8,14),(9,6),(9,7),(9,15),(10,1),(10,9),(11,16),(11,19),(12,17),(12,19),(13,8),(13,19),(15,2),(15,11),(15,12),(16,4),(16,18),(17,3),(17,18),(19,14),(19,18)],20) => ([],1)
['A',6] => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21) => ([(0,9),(0,10),(3,7),(3,14),(4,8),(4,15),(5,3),(5,16),(6,4),(6,17),(7,2),(7,12),(8,1),(8,13),(9,5),(9,11),(10,6),(10,11),(11,16),(11,17),(14,12),(14,19),(15,13),(15,20),(16,14),(16,18),(17,15),(17,18),(18,19),(18,20)],21) => ([],1)
['B',6] => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36) => ([(0,3),(3,4),(3,8),(4,20),(5,23),(6,21),(7,22),(8,12),(8,20),(9,2),(9,19),(10,13),(10,30),(11,9),(11,35),(12,10),(12,34),(13,15),(13,33),(14,11),(14,31),(15,14),(15,32),(20,7),(20,34),(21,5),(21,26),(22,6),(22,27),(23,1),(23,18),(24,25),(24,26),(25,28),(25,29),(26,23),(26,28),(27,21),(27,24),(28,17),(28,18),(29,16),(29,17),(30,27),(30,33),(31,29),(31,35),(32,25),(32,31),(33,24),(33,32),(34,22),(34,30),(35,16),(35,19)],36) => ([],1)
['C',6] => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36) => ([(0,3),(3,4),(3,8),(4,20),(5,23),(6,21),(7,22),(8,12),(8,20),(9,2),(9,19),(10,13),(10,30),(11,9),(11,35),(12,10),(12,34),(13,15),(13,33),(14,11),(14,31),(15,14),(15,32),(20,7),(20,34),(21,5),(21,26),(22,6),(22,27),(23,1),(23,18),(24,25),(24,26),(25,28),(25,29),(26,23),(26,28),(27,21),(27,24),(28,17),(28,18),(29,16),(29,17),(30,27),(30,33),(31,29),(31,35),(32,25),(32,31),(33,24),(33,32),(34,22),(34,30),(35,16),(35,19)],36) => ([],1)
['D',6] => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30) => ([(0,13),(2,25),(2,26),(5,23),(6,24),(7,14),(7,21),(8,14),(8,22),(9,12),(9,23),(10,1),(10,18),(11,10),(11,16),(12,7),(12,8),(12,15),(13,5),(13,9),(14,11),(14,28),(15,21),(15,22),(15,24),(16,17),(16,18),(19,25),(19,29),(20,26),(20,29),(21,19),(21,28),(22,20),(22,28),(23,6),(23,15),(24,2),(24,19),(24,20),(25,4),(25,27),(26,3),(26,27),(28,16),(28,29),(29,17),(29,27)],30) => ([],1)
['E',6] => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36) => ([(0,14),(2,24),(3,23),(6,16),(6,25),(7,17),(7,26),(8,19),(8,20),(9,3),(9,18),(10,2),(10,18),(11,8),(11,28),(11,29),(12,4),(12,21),(13,5),(13,22),(14,15),(15,9),(15,10),(16,30),(16,33),(17,30),(17,34),(18,11),(18,23),(18,24),(19,25),(19,35),(20,26),(20,35),(23,27),(23,28),(24,27),(24,29),(25,12),(25,33),(26,13),(26,34),(27,32),(28,6),(28,19),(28,32),(29,7),(29,20),(29,32),(30,1),(30,31),(32,16),(32,17),(32,35),(33,21),(33,31),(34,22),(34,31),(35,33),(35,34)],36) => ([],1)
Map
to root poset
Description
The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where $\alpha \prec \beta$ if $\beta - \alpha$ is a simple root.
This is the poset on the set of positive roots of its root system where $\alpha \prec \beta$ if $\beta - \alpha$ is a simple root.
Map
dual poset
Description
The dual of a poset.
The dual (or opposite) of a poset $(\mathcal P,\leq)$ is the poset $(\mathcal P^d,\leq_d)$ with $x \leq_d y$ if $y \leq x$.
The dual (or opposite) of a poset $(\mathcal P,\leq)$ is the poset $(\mathcal P^d,\leq_d)$ with $x \leq_d y$ if $y \leq x$.
Map
antichains of maximal size
Description
The lattice of antichains of maximal size in a poset.
The set of antichains of maximal size can be ordered by setting $A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B$, where $\mathop{\downarrow} A$ is the order ideal generated by $A$.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
The set of antichains of maximal size can be ordered by setting $A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B$, where $\mathop{\downarrow} A$ is the order ideal generated by $A$.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
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