Reflection group counting and q-counting

Vic Reiner Univ. of Minnesota [email protected]

Summer School on Algebraic and Enumerative Combinatorics S. Miguel de Seide, Portugal July 2-13, 2012

V. Reiner Reflection group counting and q-counting Outline

1 Lecture 1 Things we count What is a finite reflection group? Taxonomy of reflection groups 2 Lecture 2 Back to the Twelvefold Way Transitive actions and CSPs 3 Lecture 3 Multinomials, flags, and parabolic subgroups Fake degrees 4 Lecture 4 The Catalan and parking function family 5 Bibliography

V. Reiner Reflection group counting and q-counting The twelve-fold way again

balls N boxes X any f injective f surjective f n dist. dist. x (x)(x − 1)(x − 2) ··· (x − (n − 1)) x! S(n,x) x+n−1 x n−1 indist. dist. n n n−x S(n,1)


+S(n,2) 1 if n≤x dist. indist. +··· 0 else S(n,x) +S(n,x) p1(n) +p2(n) 1 if n≤x indist. indist. +··· 0 else px (n) +px (n)

V. Reiner Reflection group counting and q-counting Set partitions, number partitions, compositions

Let’s begin our reflection group generalizations with 1 Stirling numbers S(n, k) of the 2nd kind, 2 Stirling numbers s(n, k) of the 1st kind, 3 Signless Stirling numbers c(n, k) of the 1st kind, 4 Composition numbers 2n−1, 5 Partition numbers p(n).

V. Reiner Reflection group counting and q-counting Stirling numbers of the 2nd kind

Stirling numbers of the 2nd kind S(n,k) count set partitions of {1, 2,..., n} with k blocks.

These are rank numbers of the lattice Πn of set partitions partially ordered via refinement:

1234 S(4, 1) = 1 kk

V. Reiner Reflection group counting and q-counting Stirling numbers of the 2nd kind

Stirling numbers of the 2nd kind S(n,k) count set partitions of {1, 2,..., n} with k blocks.

These are rank numbers of the lattice Πn of set partitions partially ordered via refinement:

1234 S(4, 1) = 1 kk

V. Reiner Reflection group counting and q-counting Partition lattice generalizes to intersection lattice

This generalizes for any complex reflection group W to the lattice LW of all intersection subspaces of the reflecting hyperplanes, ordered via reverse inclusion,a geometric lattice.

1|234

1|2|34

1|23|4

123|4 12|34 12|3|4

V. Reiner Reflection group counting and q-counting Stirling numbers of the 1st kind

The Stirling numbers of the 1st kind s(n,k) are rank sums of Möbius function values µ(0ˆ, x) in the partition lattice Πn:

1234 −6 s(4, 1) = −6 g F SWW ggggkgkk x F SSWSWWWW ggggkkk xx FF SSSWWWW gggg kkk xx FF SSS WWWW ggggg kkk xx F SS WWWW 123|4 +2 124|3 +2 Y 134|2 +2 1|234 +2 12|34 +1 14|23 +1 13|24 +1 s(4, 2) = +11 F RR WWYWYWYYY F SS eeeeSeS ddddddd FF xRxRR xx WWWYWYFeYFeYeYSeYSeSe xSxSSddddddxdx yy xFxF RRR xx eeeeeWWFWFWWYYSYSdYSdYdxdxddd SSS xx yy x F eeRexReRee ddddFddWdWWWSxSYSYYYYY SxSS y xx eeeFee xx ddRddddd F xxWWS YxYxYY S yy 12|3|4 −1 W 13|2|4 −1 1|23|4 −1 1|2|34 −1 14|2|3 −1 1|24|3 −1 s(4, 3) = −6 WWWWW SSS FF x kk WWWW SSSS FF xx kkk WWWW SSS FF xx kkk WWWWSS F xxkkkk 1|2|3|4 +1 s(4, 4) = +1

V. Reiner Reflection group counting and q-counting A theorem of Orlik and Solomon

Theorem (Orlik-Solomon 1980)

The intersection lattice LW for a real reflection group W with degrees d1,..., dn has

n dim X µ(0ˆ, X)x = (x − (di − 1)) .

XX∈LW Yi=1

For any complex reflection group W , the same holds replacing

the exponents di − 1 with ∗ the coexponents di + 1 to be explained later.

V. Reiner Reflection group counting and q-counting A theorem of Orlik and Solomon

Theorem (Orlik-Solomon 1980)

The intersection lattice LW for a real reflection group W with degrees d1,..., dn has

n dim X µ(0ˆ, X)x = (x − (di − 1)) .

XX∈LW Yi=1

For any complex reflection group W , the same holds replacing

the exponents di − 1 with ∗ the coexponents di + 1 to be explained later.

V. Reiner Reflection group counting and q-counting Example: W = Sn

Example n s(n, n − k)x k = (x − 0)(x − 1)(x − 2) · · · (x − (n − 1)) Xk=1 +1x 4−6x 3+11x 2−6x 1 = (x − 0)(x − 1)(x − 2)(x − 3) for n = 4.

1234 −6 s(4, 1) = −6 gg F SWW ggggkkk x F SSWSWWWW ggggkkk xx FF SSSWWWW gggg kkk xx FF SSS WWWW ggggg kkk xx F SS WWWW 123|4 +2 124|3 +2 Y 134|2 +2 1|234 +2 12|34 +1 14|23 +1 13|24 +1 s(4, 2) = +11 F RR WWYWYWYYY F SS eeeeSeS ddddddd FF xRxRR xx WWWYWYFeYFeYeYSeYSeSe xSxSSddddddxdx yy FxF RRR x eeeeeWWFWFWYYSYSdYSdYdxdxddd SSS x y xx F eeRexRexRee ddddFdWdWdWWWSxSYSYYYYY SxSxS yy xx eeeFee xx ddRddddd F xxWWS YxYxYY S yy 12|3|4 −1 W 13|2|4 −1 1|23|4 −1 1|2|34 −1 14|2|3 −1 1|24|3 −1 s(4, 3) = −6 WWWW SSS FF x kk WWWWW SSS FF xx kkk WWWWSSSS FF xx kkk WWWWSS F xxkkkk 1|2|3|4 +1 s(4, 4) = +1

V. Reiner Reflection group counting and q-counting Signless Stirling numbers of the 1st kind

Definition Recall the signless Stirling number of the 1st kind c(n, k)= |s(n, k)| counts permutations w in Sn with k cycles.

One has

n c(n, k)x k = (x + 0)(x + 1)(x + 2) · · · (x + n − 1). Xk=1

Theorem (Shephard-Todd 1955, Solomon 1963)

For any complex reflection group W with degrees (d1,..., dn),

n dim V w x = (x + (di − 1)) . wX∈W Yi=1

V. Reiner Reflection group counting and q-counting Signless Stirling numbers of the 1st kind

Definition Recall the signless Stirling number of the 1st kind c(n, k)= |s(n, k)| counts permutations w in Sn with k cycles.

One has

n c(n, k)x k = (x + 0)(x + 1)(x + 2) · · · (x + n − 1). Xk=1

Theorem (Shephard-Todd 1955, Solomon 1963)

For any complex reflection group W with degrees (d1,..., dn),

n dim V w x = (x + (di − 1)) . wX∈W Yi=1

V. Reiner Reflection group counting and q-counting Signless Stirling numbers of the 1st kind

There is another ranked poset relevant here:

(1234) (1243) (1324) (1342) (1423) (1432) c(4, 1) = 6 =NVTTVTVV=RNVRVVV HURRUU. kISS jjJP 0 B ¤ =wNNTTV=vTV¡VNNRRpRVpV¡HVkRkRUk.RUvUjIjISjSpSp ¡JPJPP0 B ¤ ww = NvNvN¡=TTpVpTNVkNVNkRV¡kRHVHjVvjVv.jRVRpUpRUIpUIUS¡SSJP0PBB ¤¤ ww =v=v¡p¡pNkp=Nk=k¡T¡TjNTjVNvjRVvRVHRVHp.pVVRV¡RVI¡UUUSJSJS0 PPBP ¤ ww vv p=¡kpkk jN=j¡NjvvTNTpNpT RVH.RHVV¡ RIVIRVRVVUUUJ0S0JSSBPBP ¤¤ ww vvkpk¡pk¡=j=jj¡¡v=N=vN pp NTNT.TH¡RT¡HRVRVVVIIRRVRVVU0VUJUJSSBPSPP ¤ w kvkpkp¡jjj =¡ v p=pNN  N¡.N THTRTRRVVIVIVRR0RVJVUJVUBUBSUSPP ¤ww kkkvkpvpjpjj¡ ¡=vpvpp = NNN¡ ..NNHNHTTRTR IIVV00VRRJVJVBVUSVUSUPSPP ¤w¤wkkk vpjvpjj¡¡ ¡v¡pvp=  = ¡¡N NH TRTRI VVRVRJB VVUSVUSP (123) (132) (124) (142) (134) (143) (234) (243) (12)(34) (14)(23) (13)(24) c(4, 2) = 11 QQ N N j hh ::GGQQ--<

V. Reiner Reflection group counting and q-counting The absolute length and absolute order

Who was this ranked poset having c(n, k) as rank numbers? Definition In a complex reflection group W with set of reflections T , the absolute or reflection length is

ℓT (w) := min{ℓ : w = t1t2 · · · tℓ with ti ∈ T }.

WARNING! For real reflection groups W , this is NOT the usual Coxeter group length ℓ(w) := ℓS(w) ! Example

For W = Sn, where T is the set of transpositions tij = (i, j), and S is the subset of adjacent transpositions si = (i, i + 1), one has

ℓS(w)=#{ inversions of w} w ℓT (w)= n − #{ cycles of w} = n − 1 − dim(V )

V. Reiner Reflection group counting and q-counting The absolute length and absolute order

Who was this ranked poset having c(n, k) as rank numbers? Definition In a complex reflection group W with set of reflections T , the absolute or reflection length is

ℓT (w) := min{ℓ : w = t1t2 · · · tℓ with ti ∈ T }.

WARNING! For real reflection groups W , this is NOT the usual Coxeter group length ℓ(w) := ℓS(w) ! Example

For W = Sn, where T is the set of transpositions tij = (i, j), and S is the subset of adjacent transpositions si = (i, i + 1), one has

ℓS(w)=#{ inversions of w} w ℓT (w)= n − #{ cycles of w} = n − 1 − dim(V )

V. Reiner Reflection group counting and q-counting The absolute length and absolute order

Who was this ranked poset having c(n, k) as rank numbers? Definition In a complex reflection group W with set of reflections T , the absolute or reflection length is

ℓT (w) := min{ℓ : w = t1t2 · · · tℓ with ti ∈ T }.

WARNING! For real reflection groups W , this is NOT the usual Coxeter group length ℓ(w) := ℓS(w) ! Example

For W = Sn, where T is the set of transpositions tij = (i, j), and S is the subset of adjacent transpositions si = (i, i + 1), one has

ℓS(w)=#{ inversions of w} w ℓT (w)= n − #{ cycles of w} = n − 1 − dim(V )

V. Reiner Reflection group counting and q-counting The absolute length and absolute order

Definition Define the absolute order < on W by u < w if

−1 ℓT (u)+ ℓT (u w)= ℓT (w)

i.e., when factoring w = u · v one has ℓT (u)+ ℓT (v)= ℓT (w).

Theorem (Carter 1972, Brady-Watt 2002) For real reflection groups W acting on V = Rn, the absolute order (W ,<) is a ranked poset with rank(w)= n − dim V w .

Thus the (co-)rank generating function for (W ,<) is

n (x + (di − 1)) . Yi=1

V. Reiner Reflection group counting and q-counting The absolute length and absolute order

Definition Define the absolute order < on W by u < w if

−1 ℓT (u)+ ℓT (u w)= ℓT (w)

i.e., when factoring w = u · v one has ℓT (u)+ ℓT (v)= ℓT (w).

Theorem (Carter 1972, Brady-Watt 2002) For real reflection groups W acting on V = Rn, the absolute order (W ,<) is a ranked poset with rank(w)= n − dim V w .

Thus the (co-)rank generating function for (W ,<) is

n (x + (di − 1)) . Yi=1

V. Reiner Reflection group counting and q-counting Mapping absolute order to the intersection lattice

There is also a natural order-preserving and rank-preserving poset map (W ,<) −→ LW w 7−→ V w

Theorem (Orlik-Solomon 1980) w This map w 7→ V surjects (W ,<) ։ LW for any complex reflection group W .

V. Reiner Reflection group counting and q-counting Mapping absolute order to the intersection lattice

Example w For W = Sn, this map w 7→ V sends a permutation w to the partition π of {1, 2,..., n} whose blocks are the cycles of w.

(123) (132) 123 . H . +  . HH vv  .  +  . HvHv  .  ++  ..vv HH ..  +  vv. HH .  +  vv .  HH .  +  vv .  HH .  ++ vv .  H.  (12)(3)(13)(2)(1)(23) 12|3 13|2 23|1 = + == ¡¡ +  = ¡ ++  == ¡¡ +  == ¡¡ ++  == ¡¡ +  = ¡ +  e ¡ e

V. Reiner Reflection group counting and q-counting Set partitions mod Sn are number partitions

W = Sn acts on the set partitions Πn, with quotient poset the number partitions, ordered by refinement.

1234 4 p (4) = 1 kk

V. Reiner Reflection group counting and q-counting Set partitions mod Sn are number partitions

W = Sn acts on the set partitions Πn, with quotient poset the number partitions, ordered by refinement.

1234 4 p (4) = 1 kk

V. Reiner Reflection group counting and q-counting The intersection lattice and its W -orbits

This corresponds to the quotient map of posets

LW −→ W \LW X 7−→ W .X

where W .X is the W -orbit of the hyperplane intersection X

Thus pk (n) correspond to the rank numbers of W \LW .

V. Reiner Reflection group counting and q-counting The intersection lattice and its W -orbits

This corresponds to the quotient map of posets

LW −→ W \LW X 7−→ W .X

where W .X is the W -orbit of the hyperplane intersection X

Thus pk (n) correspond to the rank numbers of W \LW .

V. Reiner Reflection group counting and q-counting Compositions to set partitions to partitions

The refinement poset on ordered compositions of n

α = (α1,...,αℓ)

is isomorphic to the Boolean algebra 2{1,2,...,n−1}.

It naturally embeds into the lattice of set partitions Πn:

{1, 2,...,α1}|{α1 + 1, α1 + 2,...,α1 + α2}| · · ·

Example α = (2, 4, 1, 2) is sent to the partition 12|3456|7|89

One can then map the set partition to the number partitions, forgetting the order in the composition.

V. Reiner Reflection group counting and q-counting Compositions to set partitions to partitions

The refinement poset on ordered compositions of n

α = (α1,...,αℓ)

is isomorphic to the Boolean algebra 2{1,2,...,n−1}.

It naturally embeds into the lattice of set partitions Πn:

{1, 2,...,α1}|{α1 + 1, α1 + 2,...,α1 + α2}| · · ·

Example α = (2, 4, 1, 2) is sent to the partition 12|3456|7|89

One can then map the set partition to the number partitions, forgetting the order in the composition.

V. Reiner Reflection group counting and q-counting Compositions to set partitions to partitions

The refinement poset on ordered compositions of n

α = (α1,...,αℓ)

is isomorphic to the Boolean algebra 2{1,2,...,n−1}.

It naturally embeds into the lattice of set partitions Πn:

{1, 2,...,α1}|{α1 + 1, α1 + 2,...,α1 + α2}| · · ·

Example α = (2, 4, 1, 2) is sent to the partition 12|3456|7|89

One can then map the set partition to the number partitions, forgetting the order in the composition.

V. Reiner Reflection group counting and q-counting Compositions to set partitions to partitions

The refinement poset on ordered compositions of n

α = (α1,...,αℓ)

is isomorphic to the Boolean algebra 2{1,2,...,n−1}.

It naturally embeds into the lattice of set partitions Πn:

{1, 2,...,α1}|{α1 + 1, α1 + 2,...,α1 + α2}| · · ·

Example α = (2, 4, 1, 2) is sent to the partition 12|3456|7|89

One can then map the set partition to the number partitions, forgetting the order in the composition.

V. Reiner Reflection group counting and q-counting Compositions to set partitions to partitions

(4) m 12349KQQ 4 0  @@ mmmss¥ 9 KKQQQ  0  @@ mmm ss ¥¥ 99 KK QQ  0  @ mmm ss ¥ 9 KKKQQQ  0  @ mmm sss ¥¥ 9 K QQ  0 (3, 1) (2, 2) (1, 3) 123|4 124|3 U 134|2 1|234 12|34 14|23 13|24 31 22 @ @ 8 J RRURUU9K iiiK ggggg 0 @@  @@  88J¦J ¦ RRU9iU9iUKiUKi K¥Kgggg ¦ § 0  @ @ ¦8¦ JJ ¦¦ iiiRR9RUKKUgUg¥g¥gKK ¦¦ § 00   @@  @@ ¦ 88 iJi¦Jii gg99gRgRRK¥KUUUUK¦K §§ 0    ¦¦ iii ¦¦ gJgggg ¥¥RK U¦¦UK §  (2, 1, 1)(1, 2, 1)(1, 1, 2) 12|3|4 R13|2|4 1|23|4 1|2|34 14|2|3 1|24|3 211 @@  RRR KK 99 ¥ s @  RRRKK 99 ¥ ss @@  RRRKKK9 ¥¥ ss @  RRK9 ¥¥sss (1, 1, 1, 1) 1|2|3|4 1111

V. Reiner Reflection group counting and q-counting Compositions are subsets of simple reflections

The composition poset 2{1,2,...,n−1} generalizes for a real reflection groups W , with simple reflections S, to the Boolean algebra 2S. Mapping compositions → set partitions → partitions corresponds to

S 2 −→ LW −→ W \LW

J s J 7−→ V := s∈J V T X 7−→ W .X

V. Reiner Reflection group counting and q-counting Compositions are subsets of simple reflections

The composition poset 2{1,2,...,n−1} generalizes for a real reflection groups W , with simple reflections S, to the Boolean algebra 2S. Mapping compositions → set partitions → partitions corresponds to

S 2 −→ LW −→ W \LW

J s J 7−→ V := s∈J V T X 7−→ W .X

V. Reiner Reflection group counting and q-counting Ordered set partitions

The remaining entry in the last column here balls N boxes X any f injective f surjective f n dist. dist. x (x)(x − 1)(x − 2) ··· (x − (n − 1)) x! S(n,x) x+n−1 x n−1 indist. dist. n n n−x S(n,1)


+S(n,2) 1 if n≤x dist. indist. +··· 0 else S(n,x) +S(n,x) p1(n) +p2(n) 1 if n≤x indist. indist. +··· 0 else px (n) +px (n) counts something equally natural.

V. Reiner Reflection group counting and q-counting Ordered set partitions

The remaining entry in the last column here balls N boxes X any f injective f surjective f n dist. dist. x (x)(x − 1)(x − 2) ··· (x − (n − 1)) x! S(n,x) x+n−1 x n−1 indist. dist. n n n−x S(n,1)


+S(n,2) 1 if n≤x dist. indist. +··· 0 else S(n,x) +S(n,x) p1(n) +p2(n) 1 if n≤x indist. indist. +··· 0 else px (n) +px (n) counts something equally natural.

V. Reiner Reflection group counting and q-counting Ordered set partitions

Definition An ordered set partition of {1, 2,..., n} is a set partition π = (B1,..., Bℓ) with a linear ordering among the blocks Bi ,

Example ({2, 5, 6}, {1, 4}, {3, 7}) and ({3, 7}, {1, 4}, {2, 5, 6}) are different ordered set partitions of {1, 2, 3, 4, 5, 6, 7}.

There are k!S(n, k) ordered set partitions of {1, 2,..., n} with k blocks. These are the rank numbers for the refinement poset on ordered set partitions.

V. Reiner Reflection group counting and q-counting Ordered set partitions

Definition An ordered set partition of {1, 2,..., n} is a set partition π = (B1,..., Bℓ) with a linear ordering among the blocks Bi ,

Example ({2, 5, 6}, {1, 4}, {3, 7}) and ({3, 7}, {1, 4}, {2, 5, 6}) are different ordered set partitions of {1, 2, 3, 4, 5, 6, 7}.

There are k!S(n, k) ordered set partitions of {1, 2,..., n} with k blocks. These are the rank numbers for the refinement poset on ordered set partitions.

V. Reiner Reflection group counting and q-counting The geometry of ordered set partitions

They label the cones cut out by the reflecting hyperplanes. Example

({2, 5, 6}, {1, 4}, {3, 7}) ↔ {x2 = x5 = x6 ≤ x1 = x4 ≤ x3 = x7}

({3, 7}, {1, 4}, {2, 5, 6}) ↔ {x3 = x7 ≥ x1 = x4 ≥ x2 = x5 = x6}

1|234

1|2|34

1|23|4

123|4 12|34 12|3|4

Denote by ΣW the poset of all such cones ordered via inclusion.

V. Reiner Reflection group counting and q-counting The geometry of ordered set partitions

They label the cones cut out by the reflecting hyperplanes. Example

({2, 5, 6}, {1, 4}, {3, 7}) ↔ {x2 = x5 = x6 ≤ x1 = x4 ≤ x3 = x7}

({3, 7}, {1, 4}, {2, 5, 6}) ↔ {x3 = x7 ≥ x1 = x4 ≥ x2 = x5 = x6}

1|234

1|2|34

1|23|4

123|4 12|34 12|3|4

Denote by ΣW the poset of all such cones ordered via inclusion.

V. Reiner Reflection group counting and q-counting The last column of the 12-fold way is hiding a diagram

balls N boxes X surjective f ranks of refinment poset on ...

dist. dist. x! S(n,x) ordered set partitions ΣW n−1 S indist. dist. n−x compositions 2 dist. indist. S(n,x)
set partitions LW indist. indist. px (n) number partitions W \LW

S 2 −→ ΣW −→ LW −→ W \LW real and Shephard groups complex reflection groups | {z } | {z } Example {1,2}, {1,2}, (2, 3, 2) 7→ {3,4,5}, 7→ {3,4,5}, 7→ 322  {6,7}   {6,7} 

s1, x1=x2 x1=x2, x1=x2, s3,s4, 7→ ≤x3=x4=x5 7→ x3=x4=x5, 7→ S . x3=x4=x5, s x x 7 x x n 6 o  ≤x6=x7  n 6= 7 o n 6= 7 o

V. Reiner Reflection group counting and q-counting The last column of the 12-fold way is hiding a diagram

balls N boxes X surjective f ranks of refinment poset on ...

dist. dist. x! S(n,x) ordered set partitions ΣW n−1 S indist. dist. n−x compositions 2 dist. indist. S(n,x)
set partitions LW indist. indist. px (n) number partitions W \LW

S 2 −→ ΣW −→ LW −→ W \LW real and Shephard groups complex reflection groups | {z } | {z } Example {1,2}, {1,2}, (2, 3, 2) 7→ {3,4,5}, 7→ {3,4,5}, 7→ 322  {6,7}   {6,7} 

s1, x1=x2 x1=x2, x1=x2, s3,s4, 7→ ≤x3=x4=x5 7→ x3=x4=x5, 7→ S . x3=x4=x5, s x x 7 x x n 6 o  ≤x6=x7  n 6= 7 o n 6= 7 o

V. Reiner Reflection group counting and q-counting