Algebraic Combinatorics and Group Actions at Herstmonceux Castle
A broad class of shellable lattices
Russ Woodroofe
Mississippi State University
[email protected]
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Shellings
All groups and posets here are finite.
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Shellings
All groups and posets here are finite. Joint work w Jay Schweig.
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Shellings
All groups and posets here are finite. Joint work w Jay Schweig.
A shelling of a partially ordered set (or poset) P is an ordering of
the maximal chains of P satisfying certain connectivity conditions.
Not every poset has a shelling. E.g. hexagon lattice.
Those that do are called shellable.
Shellable posets have “nice” properties.
Moreover, a shelling typically exposes a
lot of deep information about the poset
in an easy-to-work-with manner.
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Shellings: importance
Shelling is “connected” ordering of maximal chains of posets,
which Russ refused to precisely define despite his talk’s title.
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Shellings: importance
Shelling is “connected” ordering of maximal chains of posets,
which Russ refused to precisely define despite his talk’s title.
So why should we care about shellings?
My favorite reason:
- (Shareshian, 2001) showed that a finite group G is solvable if and
only if its lattice of subgroups is shellable.
Also, the information exposed by a shelling is useful:
- (Adiprasito+Huh+Katz, 2016+) used shelling info in their proof
that the characteristic polynomial of any matroid is log-concave.
- (Björner+Lovasz+Yao, 1992) used shelling info in a lower bound
on complexity for k-equal partition.
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Shellings: importance
Shelling is “connected” ordering of maximal chains of posets,
which Russ refused to precisely define despite his talk’s title.
So why should we care about shellings?
My favorite reason:
- (Shareshian, 2001) showed that a finite group G is solvable if and
only if its lattice of subgroups is shellable.
Also, the information exposed by a shelling is useful:
- (Adiprasito+Huh+Katz, 2016+) used shelling info in their proof
that the characteristic polynomial of any matroid is log-concave.
- (Björner+Lovasz+Yao, 1992) used shelling info in a lower bound
on complexity for k-equal partition.
Our “broad class” of lattice includes all these classes and more.
We shell them all with a single uncomplicated proof.
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Lattices and left-modularity
Shelling is “connected” ordering of maximal chains of posets,
which appears like it might be important and useful.
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Lattices and left-modularity
Shelling is “connected” ordering of maximal chains of posets,
which appears like it might be important and useful.
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Lattices and left-modularity
Shelling is “connected” ordering of maximal chains of posets,
which appears like it might be important and useful.
A lattice is a poset with an intersection-like operation ∧
and a union-like operation ∨.
Example: Subgroup lattice of G , L(G ) =all sgs of G .
Definition: A lattice element m is
left-modular , if it never appears as the short
side of a pentagon sublattice.
y
x
← not m
Facts: 1) Any normal subgroup is l.m. in sg lattice.
2) But a l.m. subgroup need not be normal.
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Lattice classes
D: Left-modular element is lattice-theoretic analogue of normal sg.
Many important classes of lattices are defined by replacing “normal”
with “left-modular” in a definition of a class of groups.
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Lattice classes
D: Left-modular element is lattice-theoretic analogue of normal sg.
Many important classes of lattices are defined by replacing “normal”
with “left-modular” in a definition of a class of groups.
Group class
Lattice class
Characterizes
group class?
Self-dual?
abelian,
Hamiltonian
modular
No
Yes
nilpotent
dual semimodular
No
No
supersolvable
supersolvable
Yes
Yes
solvable
???
Should!!
?
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Lattices and solvability: bad examples
D: Left-modular element is lattice-theoretic analogue of normal sg.
Note: Some lattices with unpleasant properties have left-modular
analogue of composition series or chief series.
E.g.: I’d like every open interval to be connected. (at least!)
Example:
1
M
C
0
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Lattices on groups: nilpotent to solvable
D: Left-modular element is lattice-theoretic analogue of normal sg.
The correspondence Nilpotent ←→ Dual semimodular guides us:
A group is nilpotent if every maximal sg is normal
(and the same holds for every sg).
A lattice is dual semimodular if every interval [x, y ] has
every coatom m l y to be left-modular in [x, y ].
...............................................................
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Lattices on groups: nilpotent to solvable
D: Left-modular element is lattice-theoretic analogue of normal sg.
The correspondence Nilpotent ←→ Dual semimodular guides us:
A group is nilpotent if every maximal sg is normal
(and the same holds for every sg).
A lattice is dual semimodular if every interval [x, y ] has
every coatom m l y to be left-modular in [x, y ].
...............................................................
A group is solvable if some maximal sg is normal
(and the same holds for every sg).
A lattice is comodernistic (dual modernistic) if every interval [x, y ]
has some coatom m l y s.t. m is left-modular in [x, y ].
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Why “comodernistic”
D: Left-modular element is lattice-theoretic analogue of normal sg.
D: Comodernistic lattice has l.m. coatom on every interval.
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Why “comodernistic”
D: Left-modular element is lattice-theoretic analogue of normal sg.
D: Comodernistic lattice has l.m. coatom on every interval.
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Results on comodernistic lattices
D: Left-modular element is lattice-theoretic analogue of normal sg.
D: Comodernistic lattice has l.m. coatom on every interval.
Theorem 1 (us): G solvable ⇐⇒ L(G ) comodernistic.
Theorem 2 (us): L comodernistic =⇒ L shellable.
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Results on comodernistic lattices
D: Left-modular element is lattice-theoretic analogue of normal sg.
D: Comodernistic lattice has l.m. coatom on every interval.
Theorem 1 (us): G solvable ⇐⇒ L(G ) comodernistic.
Theorem 2 (us): L comodernistic =⇒ L shellable. (CL-shellable)
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Results on comodernistic lattices
D: Left-modular element is lattice-theoretic analogue of normal sg.
D: Comodernistic lattice has l.m. coatom on every interval.
Theorem 1 (us): G solvable ⇐⇒ L(G ) comodernistic.
Theorem 2 (us): L comodernistic =⇒ L shellable. (CL-shellable)
Theorem 3 (us): Many natural examples of comodernistic lattices:
1. (dual semimodular, supersolvable)
2. order congruence lattices (our main motivation)
3. k-equal partition lattices (first shelled by Björner+Welker)
4. h, k-equal partition lattices (first shelled by Björner+Sagan)
... and generalizations.
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Results on comodernistic lattices
D: Left-modular element is lattice-theoretic analogue of normal sg.
D: Comodernistic lattice has l.m. coatom on every interval.
Theorem 1 (us): G solvable ⇐⇒ L(G ) comodernistic.
Theorem 2 (us): L comodernistic =⇒ L shellable. (CL-shellable)
Theorem 3 (us): Many natural examples of comodernistic lattices:
1. (dual semimodular, supersolvable)
2. order congruence lattices (our main motivation)
3. k-equal partition lattices (first shelled by Björner+Welker)
4. h, k-equal partition lattices (first shelled by Björner+Sagan)
... and generalizations.
Question: Are Coxeter non-crossing partition lattices
comodernistic?
(Note that type A is supersolvable.)
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How our labeling works
D: Left-modular element is short side of no pentagon sublattice.
D: Comodernistic lattice has l.m. coatom on every interval.
Definition: An EL-labeling of a poset P
is a labeling
of the Hasse diagram of P s.t. ∀ intervals [x, y ]
1. There is exactly one maximal chain on [x, y ] with rising labels
(read bottom to top).
2. The rising chain may be found greedily, by starting at x and
repeatedly choosing the edge up with the least label.
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How our labeling works
D: Left-modular element is short side of no pentagon sublattice.
D: Comodernistic lattice has l.m. coatom on every interval.
Definition: An EL-labeling of a poset P (CL-labeling)
is a labeling (*) of the Hasse diagram of P s.t. ∀ intervals [x, y ]
1. There is exactly one maximal chain on [x, y ] with rising labels
(read bottom to top).
2. The rising chain may be found greedily, by starting at x and
repeatedly choosing the edge up with the least label.
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How our labeling works
D: Left-modular element is short side of no pentagon sublattice.
D: Comodernistic lattice has l.m. coatom on every interval.
Definition: An EL-labeling of a poset P (CL-labeling)
is a labeling (*) of the Hasse diagram of P s.t. ∀ intervals [x, y ]
1. There is exactly one maximal chain on [x, y ] with rising labels
(read bottom to top).
2. The rising chain may be found greedily, by starting at x and
repeatedly choosing the edge up with the least label.
Our construction:
Choose a chain 0̂ = m0 l m1 l m2 l · · · l mn = 1̂ so that
each mi is l.m. in [0̂, mi+1 ]. (analogue of composition series.)
Label atomic edge 0̂ l a with least i so that a < mi .
Recursively pick a new l.m. chain and repeat. (The hard part!)
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Jay Schweig and Russ Woodroofe, A broad class of shellable
lattices, arXiv:1604.03115.
See also:
Alireza Abdollahi, Russ Woodroofe, and Gjergji Zaimi, Frankl’s
Conjecture for subgroup lattices, arXiv:1606.00011.
Jay Schweig and Russ Woodroofe, A broad class of shellable
lattices, arXiv:1604.03115.
See also:
Alireza Abdollahi, Russ Woodroofe, and Gjergji Zaimi, Frankl’s
Conjecture for subgroup lattices, arXiv:1606.00011.
Thank you!