![Page 1: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/1.jpg)
Cyclic Sieving for Generalized Non-CrossingPartitions Associated with Complex Reflection
Groups
Christian Krattenthaler
Universitat Wien
Christian Krattenthaler Cyclic Sieving
![Page 2: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/2.jpg)
Cyclic sieving (Reiner, Stanton, White)
Ingredients:
— a set M of combinatorial objects,
— a cyclic group C = 〈g〉 acting on M,
— a polynomial P(q) in q with non-negative integer coefficients.
Definition
The triple (M,C ,P) exhibits the cyclic sieving phenomenon if
|FixM(gp)| = P(
e2πip/|C |).
Christian Krattenthaler Cyclic Sieving
![Page 3: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/3.jpg)
Cyclic sieving (Reiner, Stanton, White)
Ingredients:
— a set M of combinatorial objects,
— a cyclic group C = 〈g〉 acting on M,
— a polynomial P(q) in q with non-negative integer coefficients.
Definition
The triple (M,C ,P) exhibits the cyclic sieving phenomenon if
|FixM(gp)| = P(
e2πip/|C |).
Christian Krattenthaler Cyclic Sieving
![Page 4: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/4.jpg)
Cyclic sieving (Reiner, Stanton, White)
Ingredients:
— a set M of combinatorial objects,
— a cyclic group C = 〈g〉 acting on M,
— a polynomial P(q) in q with non-negative integer coefficients.
Definition
The triple (M,C ,P) exhibits the cyclic sieving phenomenon if
|FixM(gp)| = P(
e2πip/|C |).
Christian Krattenthaler Cyclic Sieving
![Page 5: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/5.jpg)
Example:
M ={{1, 2}, {2, 3}, {3, 4}, {1, 4}, {1, 3}, {2, 4}
}g : i 7→ i + 1 (mod 4)
P(q) =
[42
]q
= 1 + q + 2q2 + q3 + q4
|FixM(g0)| = 6 = P(1) = P(
e2πi ·0/4),
|FixM(g1)| = 0 = P(i) = P(
e2πi ·1/4),
|FixM(g2)| = 2 = P(−1) = P(
e2πi ·2/4),
|FixM(g3)| = 0 = P(−i) = P(
e2πi ·3/4).
Christian Krattenthaler Cyclic Sieving
![Page 6: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/6.jpg)
Example:
M ={{1, 2}, {2, 3}, {3, 4}, {1, 4}, {1, 3}, {2, 4}
}g : i 7→ i + 1 (mod 4)
P(q) =
[42
]q
= 1 + q + 2q2 + q3 + q4
|FixM(g0)| = 6 = P(1) = P(
e2πi ·0/4),
|FixM(g1)| = 0 = P(i) = P(
e2πi ·1/4),
|FixM(g2)| = 2 = P(−1) = P(
e2πi ·2/4),
|FixM(g3)| = 0 = P(−i) = P(
e2πi ·3/4).
Christian Krattenthaler Cyclic Sieving
![Page 7: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/7.jpg)
Example:
M ={{1, 2}, {2, 3}, {3, 4}, {1, 4}, {1, 3}, {2, 4}
}g : i 7→ i + 1 (mod 4)
P(q) =
[42
]q
= 1 + q + 2q2 + q3 + q4
|FixM(g0)| = 6 = P(1) = P(
e2πi ·0/4),
|FixM(g1)| = 0 = P(i) = P(
e2πi ·1/4),
|FixM(g2)| = 2 = P(−1) = P(
e2πi ·2/4),
|FixM(g3)| = 0 = P(−i) = P(
e2πi ·3/4).
Christian Krattenthaler Cyclic Sieving
![Page 8: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/8.jpg)
Example:
M ={{1, 2}, {2, 3}, {3, 4}, {1, 4}, {1, 3}, {2, 4}
}g : i 7→ i + 1 (mod 4)
P(q) =
[42
]q
= 1 + q + 2q2 + q3 + q4
|FixM(g0)| = 6 = P(1) = P(
e2πi ·0/4),
|FixM(g1)| = 0 = P(i) = P(
e2πi ·1/4),
|FixM(g2)| = 2 = P(−1) = P(
e2πi ·2/4),
|FixM(g3)| = 0 = P(−i) = P(
e2πi ·3/4).
Christian Krattenthaler Cyclic Sieving
![Page 9: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/9.jpg)
Example:
M ={{1, 2}, {2, 3}, {3, 4}, {1, 4}, {1, 3}, {2, 4}
}g : i 7→ i + 1 (mod 4)
P(q) =
[42
]q
= 1 + q + 2q2 + q3 + q4
|FixM(g0)| = 6 = P(1) = P(
e2πi ·0/4),
|FixM(g1)| = 0 = P(i) = P(
e2πi ·1/4),
|FixM(g2)| = 2 = P(−1) = P(
e2πi ·2/4),
|FixM(g3)| = 0 = P(−i) = P(
e2πi ·3/4).
Christian Krattenthaler Cyclic Sieving
![Page 10: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/10.jpg)
Example:
M ={{1, 2}, {2, 3}, {3, 4}, {1, 4}, {1, 3}, {2, 4}
}g : i 7→ i + 1 (mod 4)
P(q) =
[42
]q
= 1 + q + 2q2 + q3 + q4
|FixM(g0)| = 6 = P(1) = P(
e2πi ·0/4),
|FixM(g1)| = 0 = P(i) = P(
e2πi ·1/4),
|FixM(g2)| = 2 = P(−1) = P(
e2πi ·2/4),
|FixM(g3)| = 0 = P(−i) = P(
e2πi ·3/4).
Christian Krattenthaler Cyclic Sieving
![Page 11: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/11.jpg)
Cyclic sieving: equivalent characterisations
Fact
The triple (M,C ,P) exhibits the cyclic sieving phenomenon if andonly if
P(q) ≡|C |−1∑j=0
ajqj mod q|C | − 1,
where aj is the number of C -orbits for which the stabilizer orderdivides j.
Fact
Let g be a generator of the cyclic group C , and let V (j) denote the(one-dimensional) irreducible representation of C given byg · v = e2πi j/|C |v. Furthermore, let P(q) =
∑j≥0 pjq
j . Then thetriple (M,C ,P) exhibits the cyclic sieving phenomenon if and onlyif CM is isomorphic to
⊕j≥0 pjV
(j).
Christian Krattenthaler Cyclic Sieving
![Page 12: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/12.jpg)
Cyclic sieving: equivalent characterisations
Fact
The triple (M,C ,P) exhibits the cyclic sieving phenomenon if andonly if
P(q) ≡|C |−1∑j=0
ajqj mod q|C | − 1,
where aj is the number of C -orbits for which the stabilizer orderdivides j.
Fact
Let g be a generator of the cyclic group C , and let V (j) denote the(one-dimensional) irreducible representation of C given byg · v = e2πi j/|C |v. Furthermore, let P(q) =
∑j≥0 pjq
j . Then thetriple (M,C ,P) exhibits the cyclic sieving phenomenon if and onlyif CM is isomorphic to
⊕j≥0 pjV
(j).
Christian Krattenthaler Cyclic Sieving
![Page 13: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/13.jpg)
Cyclic sieving: equivalent characterisations
Fact
The triple (M,C ,P) exhibits the cyclic sieving phenomenon if andonly if
P(q) ≡|C |−1∑j=0
ajqj mod q|C | − 1,
where aj is the number of C -orbits for which the stabilizer orderdivides j.
Fact
Let g be a generator of the cyclic group C , and let V (j) denote the(one-dimensional) irreducible representation of C given byg · v = e2πi j/|C |v. Furthermore, let P(q) =
∑j≥0 pjq
j . Then thetriple (M,C ,P) exhibits the cyclic sieving phenomenon if and onlyif CM is isomorphic to
⊕j≥0 pjV
(j).
Christian Krattenthaler Cyclic Sieving
![Page 14: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/14.jpg)
History of Cyclic sieving
early 1990s: “(−1)-phenomenon” for plane partitions (JohnStembridge)
2004: “The cyclic sieving phenomenon” (Vic Reiner, DennisStanton, Dennis White)
Instances of cylic sieving were discovered for permutations, fortableaux, for non-crossing matchings, for non-crossingpartitions, for triangulations, for dissections of polygons, forclusters, for faces in the cluster complex, . . .
Christian Krattenthaler Cyclic Sieving
![Page 15: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/15.jpg)
History of Cyclic sieving
early 1990s: “(−1)-phenomenon” for plane partitions (JohnStembridge)
2004: “The cyclic sieving phenomenon” (Vic Reiner, DennisStanton, Dennis White)
Instances of cylic sieving were discovered for permutations, fortableaux, for non-crossing matchings, for non-crossingpartitions, for triangulations, for dissections of polygons, forclusters, for faces in the cluster complex, . . .
Christian Krattenthaler Cyclic Sieving
![Page 16: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/16.jpg)
Non-crossing partitions (Kreweras)
16
5
4 3
2
1
A non-crossing partition of {1, 2, 3, 4, 5, 6}
Christian Krattenthaler Cyclic Sieving
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Non-crossing partitions (Kreweras)
The non-crossing partitions of {1, 2, . . . , n}, say NC (n), can be(partially) ordered by refinement.
• NC (n) is a ranked poset.
• NC (n) is in fact a lattice.
• NC (n) is self-dual (→ Kreweras complement).
• |NC (n)| =1
n + 1
(2n
n
).
• There exist nice formulae for Mobius function, zetapolynomial, . . .
Christian Krattenthaler Cyclic Sieving
![Page 18: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/18.jpg)
Non-crossing partitions (Kreweras)
The non-crossing partitions of {1, 2, . . . , n}, say NC (n), can be(partially) ordered by refinement.
• NC (n) is a ranked poset.
• NC (n) is in fact a lattice.
• NC (n) is self-dual (→ Kreweras complement).
• |NC (n)| =1
n + 1
(2n
n
).
• There exist nice formulae for Mobius function, zetapolynomial, . . .
Christian Krattenthaler Cyclic Sieving
![Page 19: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/19.jpg)
Non-crossing partitions (Kreweras)
The non-crossing partitions of {1, 2, . . . , n}, say NC (n), can be(partially) ordered by refinement.
• NC (n) is a ranked poset.
• NC (n) is in fact a lattice.
• NC (n) is self-dual (→ Kreweras complement).
• |NC (n)| =1
n + 1
(2n
n
).
• There exist nice formulae for Mobius function, zetapolynomial, . . .
Christian Krattenthaler Cyclic Sieving
![Page 20: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/20.jpg)
m-divisible non-crossing partitions (Edelman)
12
3
4
5
6
7
8
9
101112
13
14
15
16
17
18
19
20
21
1
A 3-divisible non-crossing partition of {1, 2, . . . , 21}Christian Krattenthaler Cyclic Sieving
![Page 21: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/21.jpg)
m-divisible non-crossing partitions (Edelman)
The m-divisible non-crossing partitions of {1, 2, . . . ,mn}, sayNCm(n), can again be (partially) ordered by refinement.
• NCm(n) is a ranked poset.• NCm(n) is a join-semilattice.
• |NCm(n)| =1
n
((m + 1)n
n − 1
).
• There exist nice formulae for Mobius function, zetapolynomial, . . .• In particular, the number of elements of NCm(n) all block
sizes of which are equal to m is
1
n
(mn
n − 1
).
Christian Krattenthaler Cyclic Sieving
![Page 22: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/22.jpg)
m-divisible non-crossing partitions (Edelman)
The m-divisible non-crossing partitions of {1, 2, . . . ,mn}, sayNCm(n), can again be (partially) ordered by refinement.
• NCm(n) is a ranked poset.• NCm(n) is a join-semilattice.
• |NCm(n)| =1
n
((m + 1)n
n − 1
).
• There exist nice formulae for Mobius function, zetapolynomial, . . .• In particular, the number of elements of NCm(n) all block
sizes of which are equal to m is
1
n
(mn
n − 1
).
Christian Krattenthaler Cyclic Sieving
![Page 23: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/23.jpg)
m-divisible non-crossing partitions (Edelman)
The m-divisible non-crossing partitions of {1, 2, . . . ,mn}, sayNCm(n), can again be (partially) ordered by refinement.
• NCm(n) is a ranked poset.• NCm(n) is a join-semilattice.
• |NCm(n)| =1
n
((m + 1)n
n − 1
).
• There exist nice formulae for Mobius function, zetapolynomial, . . .• In particular, the number of elements of NCm(n) all block
sizes of which are equal to m is
1
n
(mn
n − 1
).
Christian Krattenthaler Cyclic Sieving
![Page 24: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/24.jpg)
A cyclic action: rotation
12
3
4
5
6
7
8
9
101112
13
14
15
16
17
18
19
20
21
1
Christian Krattenthaler Cyclic Sieving
![Page 25: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/25.jpg)
A cyclic action: rotation
12
3
4
5
6
7
8
9
101112
13
14
15
16
17
18
19
20
21
1
Christian Krattenthaler Cyclic Sieving
![Page 26: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/26.jpg)
A cyclic action: rotation
12
3
4
5
6
7
8
9
101112
13
14
15
16
17
18
19
20
21
Christian Krattenthaler Cyclic Sieving
![Page 27: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/27.jpg)
A cyclic action: rotation
12
3
4
5
6
7
8
9
101112
13
14
15
16
17
18
19
20
21
Christian Krattenthaler Cyclic Sieving
![Page 28: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/28.jpg)
A cyclic action: rotation
12
3
4
5
6
7
8
9
101112
13
14
15
16
17
18
19
20
21
Christian Krattenthaler Cyclic Sieving
![Page 29: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/29.jpg)
Non-crossing partitions and cyclic sieving I
Take:
— M = m-divisible non-crossing partitions of {1, 2, . . . ,mn},— C = 〈rotation〉,
— P(q) =1
[n]q
[(m + 1)n
n − 1
]q
.
Claim: The triple (M,C ,P) exhibits the cyclic sievingphenomenon.
Christian Krattenthaler Cyclic Sieving
![Page 30: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/30.jpg)
Non-crossing partitions and cyclic sieving ITake:
— M = m-divisible non-crossing partitions of {1, 2, . . . ,mn},— C = 〈rotation〉,
— P(q) =1
[n]q
[(m + 1)n
n − 1
]q
.
Claim: The triple (M,C ,P) exhibits the cyclic sievingphenomenon.
Christian Krattenthaler Cyclic Sieving
![Page 31: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/31.jpg)
Non-crossing partitions and cyclic sieving ITake:
— M = m-divisible non-crossing partitions of {1, 2, . . . ,mn},— C = 〈rotation〉,
— P(q) =1
[n]q
[(m + 1)n
n − 1
]q
.
Claim: The triple (M,C ,P) exhibits the cyclic sievingphenomenon.
Christian Krattenthaler Cyclic Sieving
![Page 32: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/32.jpg)
Non-crossing partitions and cyclic sieving ITake:
— M = m-divisible non-crossing partitions of {1, 2, . . . ,mn},— C = 〈rotation〉,
— P(q) =1
[n]q
[(m + 1)n
n − 1
]q
.
Claim: The triple (M,C ,P) exhibits the cyclic sievingphenomenon.
Christian Krattenthaler Cyclic Sieving
![Page 33: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/33.jpg)
Non-crossing partitions and cyclic sieving II
Take:
— M = non-crossing partitions of {1, 2, . . . ,mn} all block sizesof which are equal to m,
— C = 〈rotation〉,
— P(q) =1
[n]q
[mn
n − 1
]q
.
Claim: The triple (M,C ,P) exhibits the cyclic sievingphenomenon.
Christian Krattenthaler Cyclic Sieving
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Non-crossing partitions and cyclic sieving IITake:
— M = non-crossing partitions of {1, 2, . . . ,mn} all block sizesof which are equal to m,
— C = 〈rotation〉,
— P(q) =1
[n]q
[mn
n − 1
]q
.
Claim: The triple (M,C ,P) exhibits the cyclic sievingphenomenon.
Christian Krattenthaler Cyclic Sieving
![Page 35: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/35.jpg)
Non-crossing partitions and cyclic sieving IITake:
— M = non-crossing partitions of {1, 2, . . . ,mn} all block sizesof which are equal to m,
— C = 〈rotation〉,
— P(q) =1
[n]q
[mn
n − 1
]q
.
Claim: The triple (M,C ,P) exhibits the cyclic sievingphenomenon.
Christian Krattenthaler Cyclic Sieving
![Page 36: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/36.jpg)
Non-crossing partitions and cyclic sieving IITake:
— M = non-crossing partitions of {1, 2, . . . ,mn} all block sizesof which are equal to m,
— C = 〈rotation〉,
— P(q) =1
[n]q
[mn
n − 1
]q
.
Claim: The triple (M,C ,P) exhibits the cyclic sievingphenomenon.
Christian Krattenthaler Cyclic Sieving
![Page 37: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/37.jpg)
m-divisible non-crossing partitions for complex reflection groups!
(Armstrong, Brady, Watt, Bessis)
Christian Krattenthaler Cyclic Sieving
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An algebraic point of view
Define the absolute length `T (σ) of a permutation σ ∈ Sn by thesmallest k such that
σ = t1t2 · · · tk ,
where all ti are transpositions.
Define the absolute order ≤T by
σ ≤T π if and only if `T (σ) + `T (σ−1π) = `T (π).
For example,
(1, 2, 4)(3)(5, 6) ≤T (1, 2, 3, 4, 5, 6).
Christian Krattenthaler Cyclic Sieving
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An algebraic point of view
Define the absolute length `T (σ) of a permutation σ ∈ Sn by thesmallest k such that
σ = t1t2 · · · tk ,
where all ti are transpositions.
Define the absolute order ≤T by
σ ≤T π if and only if `T (σ) + `T (σ−1π) = `T (π).
For example,
(1, 2, 4)(3)(5, 6) ≤T (1, 2, 3, 4, 5, 6).
Christian Krattenthaler Cyclic Sieving
![Page 40: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/40.jpg)
An algebraic point of view
Define the absolute length `T (σ) of a permutation σ ∈ Sn by thesmallest k such that
σ = t1t2 · · · tk ,
where all ti are transpositions.
Define the absolute order ≤T by
σ ≤T π if and only if `T (σ) + `T (σ−1π) = `T (π).
For example,
(1, 2, 4)(3)(5, 6) ≤T (1, 2, 3, 4, 5, 6).
Christian Krattenthaler Cyclic Sieving
![Page 41: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/41.jpg)
An algebraic point of view
Define the absolute length `T (σ) of a permutation σ ∈ Sn by thesmallest k such that
σ = t1t2 · · · tk ,
where all ti are transpositions.
Define the absolute order ≤T by
σ ≤T π if and only if `T (σ) + `T (σ−1π) = `T (π).
For example,
(1, 2, 4)(3)(5, 6) ≤T (1, 2, 3, 4, 5, 6).
Christian Krattenthaler Cyclic Sieving
![Page 42: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/42.jpg)
For example,(1, 2, 4)(3)(5, 6) ≤T (1, 2, 3, 4, 5, 6).
16
5
4 3
2
1
Indeed, one can show that the non-crossing partitions of{1, 2, . . . , n} are in bijection with
{σ ∈ Sn : σ ≤T (1, 2, . . . , n)}.
Christian Krattenthaler Cyclic Sieving
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For example,(1, 2, 4)(3)(5, 6) ≤T (1, 2, 3, 4, 5, 6).
16
5
4 3
2
1
Indeed, one can show that the non-crossing partitions of{1, 2, . . . , n} are in bijection with
{σ ∈ Sn : σ ≤T (1, 2, . . . , n)}.
Christian Krattenthaler Cyclic Sieving
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For example,(1, 2, 4)(3)(5, 6) ≤T (1, 2, 3, 4, 5, 6).
16
5
4 3
2
1
Indeed, one can show that the non-crossing partitions of{1, 2, . . . , n} are in bijection with
{σ ∈ Sn : σ ≤T (1, 2, . . . , n)}.
Christian Krattenthaler Cyclic Sieving
![Page 45: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/45.jpg)
Complex reflection groups
A complex reflection is a linear transformation on Cn which fixes ahyperplane pointwise, and which has finite order. In other words, acomplex reflection is a diagonalisable linear transformation on Cn
whose eigenvalues are 1 with multiplicity n − 1, and whoseremaining eigenvalue is a root of unity.
A complex reflection group W is a group generated by (complex)reflections. Here, we consider always finite complex reflectiongroups.
Christian Krattenthaler Cyclic Sieving
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Complex reflection groups
A complex reflection is a linear transformation on Cn which fixes ahyperplane pointwise, and which has finite order. In other words, acomplex reflection is a diagonalisable linear transformation on Cn
whose eigenvalues are 1 with multiplicity n − 1, and whoseremaining eigenvalue is a root of unity.
A complex reflection group W is a group generated by (complex)reflections. Here, we consider always finite complex reflectiongroups.
Christian Krattenthaler Cyclic Sieving
![Page 47: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/47.jpg)
Complex reflection groups
A complex reflection is a linear transformation on Cn which fixes ahyperplane pointwise, and which has finite order. In other words, acomplex reflection is a diagonalisable linear transformation on Cn
whose eigenvalues are 1 with multiplicity n − 1, and whoseremaining eigenvalue is a root of unity.
A complex reflection group W is a group generated by (complex)reflections.
Here, we consider always finite complex reflectiongroups.
Christian Krattenthaler Cyclic Sieving
![Page 48: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/48.jpg)
Complex reflection groups
A complex reflection is a linear transformation on Cn which fixes ahyperplane pointwise, and which has finite order. In other words, acomplex reflection is a diagonalisable linear transformation on Cn
whose eigenvalues are 1 with multiplicity n − 1, and whoseremaining eigenvalue is a root of unity.
A complex reflection group W is a group generated by (complex)reflections. Here, we consider always finite complex reflectiongroups.
Christian Krattenthaler Cyclic Sieving
![Page 49: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/49.jpg)
The classification of all finite complex reflection groups
(Shephard and Todd)
All finite complex reflection groups are known!
All irreducible finite complex reflection groups are:
— the infinite family G (d , e, n), where d , e, n are positiveintegers such that e | d ,
— the exceptional groups G4,G5, . . . ,G37.
Any finite complex reflection group is a direct product ofirreducible ones.
Christian Krattenthaler Cyclic Sieving
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The classification of all finite complex reflection groups
(Shephard and Todd)
All finite complex reflection groups are known!
All irreducible finite complex reflection groups are:
— the infinite family G (d , e, n), where d , e, n are positiveintegers such that e | d ,
— the exceptional groups G4,G5, . . . ,G37.
Any finite complex reflection group is a direct product ofirreducible ones.
Christian Krattenthaler Cyclic Sieving
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The groups G (d , e, n)
Let d , e, n be positive integers such that e | d . The groupG (d , e, n) consists of all n × n matrices, in which:
• exactly one entry in each row and in each column is non-zero;
• this non-zero entry is always some d-th root of unity;
• the product of all non-zero entries is a (d/e)-th root of unity.
Special cases:
• G (1, 1, n) = Sn.
• G (2, 1, n) = Bn.
• G (2, 2, n) = Dn.
Christian Krattenthaler Cyclic Sieving
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The groups G (d , e, n)
Let d , e, n be positive integers such that e | d . The groupG (d , e, n) consists of all n × n matrices, in which:
• exactly one entry in each row and in each column is non-zero;
• this non-zero entry is always some d-th root of unity;
• the product of all non-zero entries is a (d/e)-th root of unity.
Special cases:
• G (1, 1, n) = Sn.
• G (2, 1, n) = Bn.
• G (2, 2, n) = Dn.
Christian Krattenthaler Cyclic Sieving
![Page 53: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/53.jpg)
The groups G (d , e, n)
Let d , e, n be positive integers such that e | d . The groupG (d , e, n) consists of all n × n matrices, in which:
• exactly one entry in each row and in each column is non-zero;
• this non-zero entry is always some d-th root of unity;
• the product of all non-zero entries is a (d/e)-th root of unity.
Special cases:
• G (1, 1, n) = Sn.
• G (2, 1, n) = Bn.
• G (2, 2, n) = Dn.
Christian Krattenthaler Cyclic Sieving
![Page 54: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/54.jpg)
The groups G (d , e, n)
Let d , e, n be positive integers such that e | d . The groupG (d , e, n) consists of all n × n matrices, in which:
• exactly one entry in each row and in each column is non-zero;
• this non-zero entry is always some d-th root of unity;
• the product of all non-zero entries is a (d/e)-th root of unity.
Special cases:
• G (1, 1, n) = Sn.
• G (2, 1, n) = Bn.
• G (2, 2, n) = Dn.
Christian Krattenthaler Cyclic Sieving
![Page 55: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/55.jpg)
Well-generated complex reflection groups
A complex reflection group W of rank n is called well-generated , ifit is generated by n (complex) reflections.
Christian Krattenthaler Cyclic Sieving
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Well-generated complex reflection groups
A complex reflection group W of rank n is called well-generated , ifit is generated by n (complex) reflections.
Christian Krattenthaler Cyclic Sieving
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The classification of all well-generated complex reflection groups
(Shephard and Todd)
All irreducible well-generated complex reflection groups are:
— the two infinite families G (d , 1, n) and G (e, e, n), whered , e, n are positive integers,
— the exceptional groupsG4,G5,G6,G8,G9,G10,G14,G16,G17,G18,G20,G21,G23 = H3,G24,G25,G26,G27, G28 = F4,G29,G30 = H4,G32,G33, G34,G35 = E6, G36 = E7, G37 = E8.
Christian Krattenthaler Cyclic Sieving
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Absolute order for complex reflection groups
Given a complex reflection group W , define the absolute length`T (w) of an element w ∈W by the smallest k such that
w = t1t2 · · · tk ,
where all ti are (complex) reflections.
Define the absolute order ≤T by
u ≤T w if and only if `T (u) + `T (u−1w) = `T (w).
Christian Krattenthaler Cyclic Sieving
![Page 59: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/59.jpg)
Absolute order for complex reflection groups
Given a complex reflection group W , define the absolute length`T (w) of an element w ∈W by the smallest k such that
w = t1t2 · · · tk ,
where all ti are (complex) reflections.
Define the absolute order ≤T by
u ≤T w if and only if `T (u) + `T (u−1w) = `T (w).
Christian Krattenthaler Cyclic Sieving
![Page 60: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/60.jpg)
Absolute order for complex reflection groups
Given a complex reflection group W , define the absolute length`T (w) of an element w ∈W by the smallest k such that
w = t1t2 · · · tk ,
where all ti are (complex) reflections.
Define the absolute order ≤T by
u ≤T w if and only if `T (u) + `T (u−1w) = `T (w).
Christian Krattenthaler Cyclic Sieving
![Page 61: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/61.jpg)
Non-crossing partitions for reflection groups
The degrees d1 ≤ d2 ≤ · · · ≤ dn of a (complex) reflection group Ware the degrees of a system of homogeneous polynomial generatorsof the invariant ring of W . The largest degree, dn, is calledCoxeter number, and is denoted by h.
A regular element (in the sense of Springer) is an element w ∈Wwhich has an eigenvalue, ζ say, such that the correspondingeigenvector lies in no reflection hyperplane. If this eigenvalue ζ is aprimitive h-th root of unity, then w is called a Coxeter element.We always write c for Coxeter elements.
The non-crossing partitions for a well-generated complex reflectiongroup W are defined by
NC (W ) := {w ∈W : w ≤T c},where c is a Coxeter element in W .
Christian Krattenthaler Cyclic Sieving
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Non-crossing partitions for reflection groups
The degrees d1 ≤ d2 ≤ · · · ≤ dn of a (complex) reflection group Ware the degrees of a system of homogeneous polynomial generatorsof the invariant ring of W . The largest degree, dn, is calledCoxeter number, and is denoted by h.
A regular element (in the sense of Springer) is an element w ∈Wwhich has an eigenvalue, ζ say, such that the correspondingeigenvector lies in no reflection hyperplane. If this eigenvalue ζ is aprimitive h-th root of unity, then w is called a Coxeter element.We always write c for Coxeter elements.
The non-crossing partitions for a well-generated complex reflectiongroup W are defined by
NC (W ) := {w ∈W : w ≤T c},where c is a Coxeter element in W .
Christian Krattenthaler Cyclic Sieving
![Page 63: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/63.jpg)
Non-crossing partitions for reflection groups
The degrees d1 ≤ d2 ≤ · · · ≤ dn of a (complex) reflection group Ware the degrees of a system of homogeneous polynomial generatorsof the invariant ring of W . The largest degree, dn, is calledCoxeter number, and is denoted by h.
A regular element (in the sense of Springer) is an element w ∈Wwhich has an eigenvalue, ζ say, such that the correspondingeigenvector lies in no reflection hyperplane. If this eigenvalue ζ is aprimitive h-th root of unity, then w is called a Coxeter element.We always write c for Coxeter elements.
The non-crossing partitions for a well-generated complex reflectiongroup W are defined by
NC (W ) := {w ∈W : w ≤T c},where c is a Coxeter element in W .
Christian Krattenthaler Cyclic Sieving
![Page 64: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/64.jpg)
Non-crossing partitions for reflection groups
The degrees d1 ≤ d2 ≤ · · · ≤ dn of a (complex) reflection group Ware the degrees of a system of homogeneous polynomial generatorsof the invariant ring of W . The largest degree, dn, is calledCoxeter number, and is denoted by h.
A regular element (in the sense of Springer) is an element w ∈Wwhich has an eigenvalue, ζ say, such that the correspondingeigenvector lies in no reflection hyperplane. If this eigenvalue ζ is aprimitive h-th root of unity, then w is called a Coxeter element.We always write c for Coxeter elements.
The non-crossing partitions for a well-generated complex reflectiongroup W are defined by
NC (W ) := {w ∈W : w ≤T c},where c is a Coxeter element in W .
Christian Krattenthaler Cyclic Sieving
![Page 65: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/65.jpg)
Non-crossing partitions for reflection groups
Everything generalises to NC (W ):
— order relation: ≤T
— NC (W ) is a ranked poset:
rank of w = `T (w)
— NC (W ) is a lattice
— NC (W ) is self-dual:“Kreweras-complement” is w 7→ cw−1
— Catalan number for W : if W is irreducible then
|NC (W )| =n∏
i=1
h + di
di
Christian Krattenthaler Cyclic Sieving
![Page 66: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/66.jpg)
m-divisible non-crossing partitions for reflection groups (Armstrong)
The m-divisible non-crossing partitions for a complex reflectiongroup W are defined by
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
where c is a Coxeter element in W .
In particular,NC 1(W ) ∼= NC (W ).
Christian Krattenthaler Cyclic Sieving
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m-divisible non-crossing partitions for reflection groups (Armstrong)
The m-divisible non-crossing partitions for a complex reflectiongroup W are defined by
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
where c is a Coxeter element in W .
In particular,NC 1(W ) ∼= NC (W ).
Christian Krattenthaler Cyclic Sieving
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m-divisible non-crossing partitions for reflection groups (Armstrong)
The m-divisible non-crossing partitions for a complex reflectiongroup W are defined by
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
where c is a Coxeter element in W .
In particular,NC 1(W ) ∼= NC (W ).
Christian Krattenthaler Cyclic Sieving
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Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6(= S7):w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21) (7, 16)−1 (2, 20)−1 (3, 6, 18)−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 70: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/70.jpg)
Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6
(= S7):w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21) (7, 16)−1 (2, 20)−1 (3, 6, 18)−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 71: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/71.jpg)
Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6(= S7):
w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21) (7, 16)−1 (2, 20)−1 (3, 6, 18)−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 72: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/72.jpg)
Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6(= S7):w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).
Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21) (7, 16)−1 (2, 20)−1
(3, 6, 18)
−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 73: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/73.jpg)
Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6(= S7):w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21) (7, 16)−1
(2, 20)
−1
(3, 6, 18)
−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 74: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/74.jpg)
Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6(= S7):w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21)
(7, 16)
−1
(2, 20)
−1
(3, 6, 18)
−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 75: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/75.jpg)
Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6(= S7):w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21)
(7, 16)−1 (2, 20)−1 (3, 6, 18)−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 76: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/76.jpg)
Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6(= S7):w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21) (7, 16)−1 (2, 20)−1 (3, 6, 18)−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 77: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/77.jpg)
Combinatorial realisation in type A (Armstrong)
NCm(W ) ={
(w0; w1, . . . ,wm) : w0w1 · · ·wm = c and
`T (w0) + `T (w1) + · · ·+ `T (wm) = `T (c)},
Example for m = 3, W = A6(= S7):w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6).Now “blow-up” w1,w2,w3:
(1, 2, . . . , 21) (7, 16)−1 (2, 20)−1 (3, 6, 18)−1
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
Christian Krattenthaler Cyclic Sieving
![Page 78: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/78.jpg)
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A 3-divisible non-crossing partition of type A6
Christian Krattenthaler Cyclic Sieving
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A 3-divisible non-crossing partition of type B5
Christian Krattenthaler Cyclic Sieving
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A 3-divisible non-crossing partition of type D6
Christian Krattenthaler Cyclic Sieving
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Properties of NCm(W )
— order relation:
(u0; u1, . . . , um) ≤ (w0; w1, . . . ,wm)
if and only if u1 ≥ w1, . . . , um ≥ wm;
— NCm(W ) is a join-semilattice;
— NCm(W ) is ranked :
rank of (w0; w1, . . . ,wm) = `T (w0)
Christian Krattenthaler Cyclic Sieving
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The Fuß–Catalan numbers for reflection groups
Theorem (Athanasiadis, Bessis, Corran, Chapoton,Edelman, Reiner)
If W is irreducible then
|NCm(W )| =n∏
i=1
mh + di
di.
Christian Krattenthaler Cyclic Sieving
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Let φ : NCm(W )→ NCm(W ) be the map defined by
(w0; w1, . . . ,wm)
7→((cwmc−1)w0(cwmc−1)−1; cwmc−1,w1,w2, . . . ,wm−1
).
It generates a cyclic group of order mh.
Christian Krattenthaler Cyclic Sieving
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The action combinatorially (type A)
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Christian Krattenthaler Cyclic Sieving
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The action combinatorially (type A)
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Christian Krattenthaler Cyclic Sieving
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The action combinatorially (type A)
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Christian Krattenthaler Cyclic Sieving
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The action combinatorially (type A)
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Christian Krattenthaler Cyclic Sieving
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The action combinatorially (type A)
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Christian Krattenthaler Cyclic Sieving
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Let φ : NCm(W )→ NCm(W ) be the map defined by
(w0; w1, . . . ,wm)
7→((cwmc−1)w0(cwmc−1)−1; cwmc−1,w1,w2, . . . ,wm−1
).
It generates a cyclic group of order mh.
Furthermore, let
Catm(W ; q) :=n∏
i=1
[mh + di ]q[di ]q
,
where [α]q := (1− qα)/(1− q).
Theorem (with T. W. Muller)
The triple (NCm(W ), 〈φ〉,Catm(W ; q)) exhibits the cyclic sievingphenomenon.
(Originally conjectured by Armstrong, Bessis and Reiner)
Christian Krattenthaler Cyclic Sieving
![Page 90: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/90.jpg)
Let φ : NCm(W )→ NCm(W ) be the map defined by
(w0; w1, . . . ,wm)
7→((cwmc−1)w0(cwmc−1)−1; cwmc−1,w1,w2, . . . ,wm−1
).
It generates a cyclic group of order mh.Furthermore, let
Catm(W ; q) :=n∏
i=1
[mh + di ]q[di ]q
,
where [α]q := (1− qα)/(1− q).
Theorem (with T. W. Muller)
The triple (NCm(W ), 〈φ〉,Catm(W ; q)) exhibits the cyclic sievingphenomenon.
(Originally conjectured by Armstrong, Bessis and Reiner)
Christian Krattenthaler Cyclic Sieving
![Page 91: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/91.jpg)
Let φ : NCm(W )→ NCm(W ) be the map defined by
(w0; w1, . . . ,wm)
7→((cwmc−1)w0(cwmc−1)−1; cwmc−1,w1,w2, . . . ,wm−1
).
It generates a cyclic group of order mh.Furthermore, let
Catm(W ; q) :=n∏
i=1
[mh + di ]q[di ]q
,
where [α]q := (1− qα)/(1− q).
Theorem (with T. W. Muller)
The triple (NCm(W ), 〈φ〉,Catm(W ; q)) exhibits the cyclic sievingphenomenon.
(Originally conjectured by Armstrong, Bessis and Reiner)
Christian Krattenthaler Cyclic Sieving
![Page 92: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/92.jpg)
Let ψ : NCm(W )→ NCm(W ) be the map defined by
(w0; w1, . . . ,wm) 7→(cwmc−1; w0,w1, . . . ,wm−1
).
It generates a group of order (m + 1)h.
(If we embed
(w0; w1, . . . ,wm) 7→(id; w0,w1, . . . ,wm
).
then, in types A, B and D, we are talking about non-crossing partitions
all blocks of which have size m + 1, and this action is again rotation.)
Furthermore, let
Catm(W ; q) :=n∏
i=1
[mh + di ]q[di ]q
,
where [α]q := (1− qα)/(1− q).
Theorem (with T. W. Muller)
The triple (NCm(W ), 〈ψ〉,Catm(W ; q)) exhibits the cyclic sievingphenomenon.
(Originally conjectured by Bessis and Reiner)
Christian Krattenthaler Cyclic Sieving
![Page 93: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/93.jpg)
Let ψ : NCm(W )→ NCm(W ) be the map defined by
(w0; w1, . . . ,wm) 7→(cwmc−1; w0,w1, . . . ,wm−1
).
It generates a group of order (m + 1)h.
(If we embed
(w0; w1, . . . ,wm) 7→(id; w0,w1, . . . ,wm
).
then, in types A, B and D, we are talking about non-crossing partitions
all blocks of which have size m + 1, and this action is again rotation.)
Furthermore, let
Catm(W ; q) :=n∏
i=1
[mh + di ]q[di ]q
,
where [α]q := (1− qα)/(1− q).
Theorem (with T. W. Muller)
The triple (NCm(W ), 〈ψ〉,Catm(W ; q)) exhibits the cyclic sievingphenomenon.
(Originally conjectured by Bessis and Reiner)
Christian Krattenthaler Cyclic Sieving
![Page 94: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/94.jpg)
Let ψ : NCm(W )→ NCm(W ) be the map defined by
(w0; w1, . . . ,wm) 7→(cwmc−1; w0,w1, . . . ,wm−1
).
It generates a group of order (m + 1)h.
(If we embed
(w0; w1, . . . ,wm) 7→(id; w0,w1, . . . ,wm
).
then, in types A, B and D, we are talking about non-crossing partitions
all blocks of which have size m + 1, and this action is again rotation.)
Furthermore, let
Catm(W ; q) :=n∏
i=1
[mh + di ]q[di ]q
,
where [α]q := (1− qα)/(1− q).
Theorem (with T. W. Muller)
The triple (NCm(W ), 〈ψ〉,Catm(W ; q)) exhibits the cyclic sievingphenomenon.
(Originally conjectured by Bessis and Reiner)
Christian Krattenthaler Cyclic Sieving
![Page 95: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/95.jpg)
Let ψ : NCm(W )→ NCm(W ) be the map defined by
(w0; w1, . . . ,wm) 7→(cwmc−1; w0,w1, . . . ,wm−1
).
It generates a group of order (m + 1)h.
(If we embed
(w0; w1, . . . ,wm) 7→(id; w0,w1, . . . ,wm
).
then, in types A, B and D, we are talking about non-crossing partitions
all blocks of which have size m + 1, and this action is again rotation.)
Furthermore, let
Catm(W ; q) :=n∏
i=1
[mh + di ]q[di ]q
,
where [α]q := (1− qα)/(1− q).
Theorem (with T. W. Muller)
The triple (NCm(W ), 〈ψ〉,Catm(W ; q)) exhibits the cyclic sievingphenomenon.
(Originally conjectured by Bessis and Reiner)Christian Krattenthaler Cyclic Sieving
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The two cyclic sieving phenomena for NCm(G (d , 1, n)
)follow
from the following result.
Theorem
Let m, n, r be positive integers such that r ≥ 2 and r | mn. Fornon-negative integers b1, b2, . . . , bn, the number of m-divisiblenon-crossing partitions of {1, 2, . . . ,mn} (in the sense of Edelman)which are invariant under the rotation i 7→ i + mn
r mod mn andhave exactly rbi non-zero blocks of size mi, i = 1, 2, . . . , n, is givenby (
b1 + b2 + · · ·+ bn
b1, b2, . . . , bn
)(mn/r
b1 + b2 + · · ·+ bn
)if b1 + 2b2 + · · ·+ nbn ≤ bn/rc, and it is zero otherwise.
Christian Krattenthaler Cyclic Sieving
![Page 97: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/97.jpg)
In order to establish the cyclic sieving phenomena forNCm
(G (e, e, n)
), one proves analogous enumeration results for
m-divisible non-crossing partitions on an annulus.
For the exceptional groups, we do a (lengthy) computerverification.
Christian Krattenthaler Cyclic Sieving
![Page 98: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/98.jpg)
In order to establish the cyclic sieving phenomena forNCm
(G (e, e, n)
), one proves analogous enumeration results for
m-divisible non-crossing partitions on an annulus.
For the exceptional groups, we do a (lengthy) computerverification.
Christian Krattenthaler Cyclic Sieving
![Page 99: Cyclic Sieving for Generalized Non-Crossing Partitions ...kratt/vortrag/sieving2.pdf · Cyclic sieving (Reiner, Stanton, White) Ingredients: |a set M of combinatorial objects, |a](https://reader033.vdocument.in/reader033/viewer/2022050108/5f4649bbfbb9317616753e83/html5/thumbnails/99.jpg)
Christian Krattenthaler Cyclic Sieving