Download Reflection group counting and q-counting Vic Reiner Univ. of Minnesota

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
dist.
indist.
boxes X
dist.
dist.
dist.
indist.
indist.
indist.
any f
xn
x+n−1
n
S(n,1)
+S(n,2)
+···
+S(n,x)
p1 (n)
+p2 (n)
+···
+px (n)
V. Reiner
injective f
(x )(x − 1)(x − 2) · · · (x − (n − 1))
x
n
1 if n≤x
0 else
surjective f
x! S(n,x)
n−1
1 if n≤x
0 else
Reflection group counting and q-counting
n−x
S(n,x)
px (n)
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
kkkq <MMSSS
kkqkqqq <<<MMMSMSSSS
k
k
<< MMM SSSS
kkk qq S
kkk qq 124|3 S
1|234 h 12|34
123|4
134|2
14|23ff 13|24
V
V
f
V
S
h
M
M
M
;; MM
SVV< M hh M
ff
;; MMM hShShSVh<SV<hS<VhSMVhMVMVfMVfffMfMfMfMfff
S
V
h
f
M
M
M
;
<
V
h
f
S
f
V
h
M
M
S
M
f
V
h
;
<
f
S
V
h
M
M
M
f
S
V
h
f
V
hh fff
12|3|4 S 13|2|4
SSS MM1|23|4<< 1|2|34 14|2|3 q 1|24|3
SSS MMM <
qqq
SSS MM <<
SSSMM< qqqq
q
1|2|3|4
V. Reiner
S(4, 1) = 1
S(4, 2) = 7
S(4, 3) = 6
S(4, 4) = 1
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
kkkq <MMSSS
kkqkqqq <<<MMMSMSSSS
k
k
<< MMM SSSS
kkk qq S
kkk qq 124|3 S
1|234 h 12|34
123|4
134|2
14|23ff 13|24
V
V
f
V
S
h
M
M
M
;; MM
SVV< M hh M
ff
;; MMM hShShSVh<SV<hS<VhSMVhMVMVfMVfffMfMfMfMfff
S
V
h
f
M
M
M
;
<
V
h
f
S
f
V
h
M
M
S
M
f
V
h
;
<
f
S
V
h
M
M
M
f
S
V
h
f
V
hh fff
12|3|4 S 13|2|4
SSS MM1|23|4<< 1|2|34 14|2|3 q 1|24|3
SSS MMM <
qqq
SSS MM <<
SSSMM< qqqq
q
1|2|3|4
V. Reiner
S(4, 1) = 1
S(4, 2) = 7
S(4, 3) = 6
S(4, 4) = 1
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
V. Reiner
4
1|2
3|4
1|2|3
123|4
12|34
12|3|4
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(0̂, x) in the partition lattice Πn :
123|4
12|3|4
SWW
gg 1234 −6FFW
FFSSSWSWSWSWWWWW
ggkgkgkgkkkxxx
g
g
g
g
FF SSS WWWWW
ggg kkk xxx
SS
WWW
F
x
ggggg kkk
1|234 +2 ee 12|34 +1
14|23 d+1
124|3 +2 Y
134|2 +2
+2
Y
W
d
Y
W
WYY FYFYSYSeSeSeeeee SSSx dddddxddd dd 13|24
FFRRRxR
FFxx RRR xxxWeWeWYeWYW
yy
eWeFWeFWYYYSYSY YdYxdxdxdSdSdSdSS xxx
yy
xWSWSYSYYYYYYxSxSSS
xx FFFeeeeRxRexRexReRee ddddddFdWFdFWdWdWSdWYSdWxS
y
x
y
x
x eee x dddd
x YY
y
1|2|34 −1
14|2|3 −1
−1 W 13|2|4 −1
−1
1|24|3 −1
WWWWW SS1|23|4
F
k
F
WWWWWSSSSS FF
kk
xx
WWWWSWSS FF
xx kkkk
WWSS F xxkxkkkk
s(4, 1) = −6
+1
1|2|3|4 +1
V. Reiner
Reflection group counting and q-counting
s(4, 2) = +11
s(4, 3) = −6
s(4, 4) = +1
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
X
µ(0̂, X )x dim X =
X ∈LW
n
Y
(x − (di − 1)) .
i=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
X
µ(0̂, X )x dim X =
X ∈LW
n
Y
(x − (di − 1)) .
i=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
X
s(n, n − k)x k = (x − 0)(x − 1)(x − 2) · · · (x − (n − 1))
k =1
4
+1x −6x 3 +11x 2 −6x 1 = (x − 0)(x − 1)(x − 2)(x − 3) for n = 4.
123|4
12|3|4
SWW
gg 1234 −6FFW
FFSSSWSWSWSWWWWW
ggkgkgkgkkkxxx
g
g
g
g
FF SSS WWWWW
ggg kkk xxx
SS
WWW
F
x
ggggg kkk
134|2 +2
+2
14|23 +1
124|3 +2 Y
1|234 +2 ee 12|34 +1
Y
W
d
Y
W
d
Y
e
W Y FYFYSYSeSeSeeee SSSxS dddddxddd dd 13|24
FFRRRxR
y
FFxx RRR xxxWeWeWYeWYW
eWeFWeFWYYYSYSY YdYxdxdxddSdSdSS xxx
yy
yy
xx FFeFeeeeRxRexRexReRee ddddddFdWFdFWdWdWSdWYSdWxS
xWSWSYSYYYYYYxSYxSSS
y
x
y
x ee
x
x dddd
x Y
−1 W 13|2|4 −1
−1
1|2|34 −1
14|2|3 −1
1|24|3 −1
WWWWW SS1|23|4
F
k
F
x
k
WWWW SSSSS FF
xx kkkk
WWWWWSSS FF
WWWSS F xxkxkxkkkk
s(4, 1) = −6
+1
1|2|3|4 +1
V. Reiner
Reflection group counting and q-counting
s(4, 2) = +11
s(4, 3) = −6
s(4, 4) = +1
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
X
c(n, k)x k = (x + 0)(x + 1)(x + 2) · · · (x + n − 1).
k =1
Theorem (Shephard-Todd 1955, Solomon 1963)
For any complex reflection group W with degrees (d1 , . . . , dn ),
X
w
x dim V =
w ∈W
n
Y
(x + (di − 1)) .
i=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
X
c(n, k)x k = (x + 0)(x + 1)(x + 2) · · · (x + n − 1).
k =1
Theorem (Shephard-Todd 1955, Solomon 1963)
For any complex reflection group W with degrees (d1 , . . . , dn ),
X
w
x dim V =
w ∈W
n
Y
(x + (di − 1)) .
i=1
V. Reiner
Reflection group counting and q-counting
Signless Stirling numbers of the 1st kind
There is another ranked poset relevant here:
(1234) V
V(1324)
T(1243)
R(1342)
U UkvISI(1423)
TV VR
S jJP(1432)
N
RVV VpHVU
0B
HVRHkVRUHkVRUk..V
w=w=wN=NNTVNTVvNTvV==v
T=TVNTpNVRpTNVRVpk
Rv.jRVUjRUjIpUSIjpUSjpSpSSJPJSPJPP00PBBB
R
k
k
R
N
v
V
j
R
V
R
N
N
U
I
p
j
k
p
J
=
=
V
R
T
H
v
v
V
w
S
U
j
k
R
V
N
N
p
T
V
p
R
I
.
VR U SJ0 PPB
w vv==kpkpkkN=N=jNjjTjvNTvNTpVNpTRVHpRV
HVTHRVRVRVVRVIVIRVIRVUIRVURVUSVUJ0S0VUJ0SUJPSUJBPSBPSBPP
v pp NTHN.TR
wwww vvkpvkp
.
kpkpj=j=jjj v=N=vp
N
T
k
TRTRVTRVTRVIVIRIVRVRV0V0RVRJVUJVUJSBVUBSVUBPSUPSPSPP
vpvpp ==NNNN N..NHNTHNH
wwkwkkvkpvkpvjpkvjpjpjjj =v=v
NHNH TRTRTRII 0VVRVRVRJJVBVUVUSVUSVUPSP
N
.
p
=
=
w
N
vpp wkk vpjj (143)
(123) Q(132) N(124) N (142)
(134)
(234)
(243) (12)(34) (14)(23)h (13)(24)
::GGQQQ--<<NN ==NN .. HH HH
llo hhhhnn
== . jjj
::GGGQ
-- Q<Q<QNQNNN== NN.N.NHHH H HH ==jjj..j.jj lllholholholo
hohhhnnnn
:: GG- << QQ QN=N=N . NNNHH jjHHjj == .llhlhohoho nnn
:: -G- GG<< QQ=QN=N .N. NjNjNHjHjH HhHhlHhl=l
hlh.hoo nn
::- GG<< jQ=j=QjQ.NjQNjN hNhNHhNlHhl hll oHHo==o=o.. nnn nn
:- GjG<jjj h=h. hQ
hQNhh l NHHoo HH=n.n j
h lQNll oNo nn (12)
(13)
(14)
(23)
(24)
(34)
== ..
vvv
== .
vv
== ..
== ..
vvvv
== . vv
==.. vvv
v
e
V. Reiner
Reflection group counting and q-counting
c(4, 1) = 6
c(4, 2) = 11
c(4, 3) = 6
s(4, 4) = 1
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 = t1 t2 · · · 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 }
ℓT (w ) = n − #{ cycles of w } = n − 1 − dim(V w )
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 = t1 t2 · · · 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 }
ℓT (w ) = n − #{ cycles of w } = n − 1 − dim(V w )
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 = t1 t2 · · · 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 }
ℓT (w ) = n − #{ cycles of w } = n − 1 − dim(V w )
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
ℓT (u) + ℓT (u −1 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
Y
(x + (di − 1)) .
i=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
ℓT (u) + ℓT (u −1 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
Y
(x + (di − 1)) .
i=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)
This map w 7→ V w 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
For W = Sn , this map w 7→ V w sends a permutation w to the
partition π of {1, 2, . . . , n} whose blocks are the cycles of w .
(132)
...HHHHvvv ...
.
. vvvHHH
vvv... HHH ...
H
v
.. vv
HH ..
H
vv
(12)(3)
(13)(2)
(1)(23)
==
==
==
==
==
= (123)
e
V. Reiner
123
++
++
++
++
+
12|3
++ 13|2 23|1
++
++ ++ + e
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.
SSS
kk 1234<M
kkqkqqq <<M<MMSMSSSS
k
k
k q
<< MMMMSSSSS
kkk qq S
kkk qq 123|4
VSVS134|2
hM 12|34 f14|23
ff 13|24
f
VSVV<V<MMM1|234
h
M
f
h
;;MMM124|3 S
f
h
; MM ShShSV<hVhVhMVM f
MfMfMfff
;;; hMhMhMhMhhhfffS<f<S<fSfSVfMSVfMSVfMMVVVVMMMM hhh ffff
S
VV
1|2|34
14|2|3
1|24|3
12|3|4 S 13|2|4
SSS MM1|23|4
<
q
SSS MMM <<
qqq
SSS MM <<
SSSMM< qqqq
q
1|2|3|4
V. Reiner
4
31
33
33
33
33
33
33
22
p1 (4) = 1
p2 (4) = 2
211
p3 (4) = 1
1111
p4 (4) = 1
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.
SSS
kk 1234<M
kkqkqqq <<M<MMSMSSSS
k
k
k q
<< MMMMSSSSS
kkk qq S
kkk qq 123|4
VSVS134|2
hM 12|34 f14|23
ff 13|24
f
VSVV<V<MMM1|234
h
M
f
h
;;MMM124|3 S
f
h
; MM ShShSV<hVhVhMVM f
MfMfMfff
;;; hMhMhMhMhhhfffS<f<S<fSfSVfMSVfMSVfMMVVVVMMMM hhh ffff
S
VV
1|2|34
14|2|3
1|24|3
12|3|4 S 13|2|4
SSS MM1|23|4
<
q
SSS MMM <<
qqq
SSS MM <<
SSSMM< qqqq
q
1|2|3|4
V. Reiner
4
31
33
33
33
33
33
33
22
p1 (4) = 1
p2 (4) = 2
211
p3 (4) = 1
1111
p4 (4) = 1
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)
@@
@@


@@



(3, 1)
@@ (2, 2)@@ (1, 3)
@
@
@@@ @@@



(2, 1, 1)
1, 2)
@@ (1, 2, 1) (1,
@@


@@ 

(1, 1, 1, 1)
1234
mm 9 KQQQ
msmsmsss 99K9KKQKQKQQQ
m
m
99 KKK QQQ
mmm s Q
mmm sss 124|3 U
134|2
123|4
1|234 i 12|34
14|23
13|24
g
R
U
g
g
U
i
RRU 9UKKiii KK gggg
88JJJ
iU
88 JJ iiRiRU9R
UKUKKUgUgggKgKgK i9R
9
R
8 JJii
g9gRgRKKUU KK
ii8iii gJggggg 9 RRK UUUUK 12|3|4 R13|2|4 1|23|4 1|2|34 14|2|3 1|24|3
RRR KK 99
ss
RRR KKK 99
RRRKK 9 ssss
RRK9 ss
4
31
00
00
0
1|2|3|4
V. Reiner
Reflection group counting and q-counting
00
00
0
22
211
1111
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
2S −→
LW
J 7−→ V J :=
T
s∈J
X
V. Reiner
−→ W \LW
Vs
7−→ W .X
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
2S −→
LW
J 7−→ V J :=
T
s∈J
X
V. Reiner
−→ W \LW
Vs
7−→ W .X
Reflection group counting and q-counting
Ordered set partitions
The remaining entry in the last column here
balls N
dist.
indist.
boxes X
dist.
dist.
dist.
indist.
indist.
indist.
any f
xn
injective f
(x )(x − 1)(x − 2) · · · (x − (n − 1))
x+n−1
n
S(n,1)
+S(n,2)
+···
+S(n,x)
p1 (n)
+p2 (n)
+···
+px (n)
x
n
1 if n≤x
0 else
surjective f
x! S(n,x)
n−1
1 if n≤x
0 else
counts something equally natural.
V. Reiner
Reflection group counting and q-counting
n−x
S(n,x)
px (n)
Ordered set partitions
The remaining entry in the last column here
balls N
dist.
indist.
boxes X
dist.
dist.
dist.
indist.
indist.
indist.
any f
xn
injective f
(x )(x − 1)(x − 2) · · · (x − (n − 1))
x+n−1
n
S(n,1)
+S(n,2)
+···
+S(n,x)
p1 (n)
+p2 (n)
+···
+px (n)
x
n
1 if n≤x
0 else
surjective f
x! S(n,x)
n−1
1 if n≤x
0 else
counts something equally natural.
V. Reiner
Reflection group counting and q-counting
n−x
S(n,x)
px (n)
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
4
1|2
3|4
1|2|3
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
4
1|2
3|4
1|2|3
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
dist.
indist.
dist.
indist.
boxes X
dist.
dist.
indist.
indist.
surjective f
x! S(n,x)
n−1
ranks of refinment poset on ...
ordered set partitions ΣW
compositions 2S
set partitions LW
number partitions W \LW
n−x
S(n,x)
px (n)
2S −→ ΣW
|
{z
}
−→
real and Shephard groups
LW −→ W \LW
|
{z
}
complex reflection groups
Example
(2, 3, 2) 7→
n
s1 , o
s3 ,s4 ,
s6
7→
{1,2},
{3,4,5},
{6,7}
x1 =x2
≤x3 =x4 =x5
≤x6 =x7
7→
7→
V. Reiner
{1,2},
{3,4,5},
{6,7}
n
7→
x1 =x2 , o
x3 =x4 =x5 ,
x6 =x7
7→ S7 .
322
n
x1 =x2 , o
x3 =x4 =x5 ,
x6 =x7
Reflection group counting and q-counting
The last column of the 12-fold way is hiding a diagram
balls N
dist.
indist.
dist.
indist.
boxes X
dist.
dist.
indist.
indist.
surjective f
x! S(n,x)
n−1
ranks of refinment poset on ...
ordered set partitions ΣW
compositions 2S
set partitions LW
number partitions W \LW
n−x
S(n,x)
px (n)
2S −→ ΣW
|
{z
}
−→
real and Shephard groups
LW −→ W \LW
|
{z
}
complex reflection groups
Example
(2, 3, 2) 7→
n
s1 , o
s3 ,s4 ,
s6
7→
{1,2},
{3,4,5},
{6,7}
x1 =x2
≤x3 =x4 =x5
≤x6 =x7
7→
7→
V. Reiner
{1,2},
{3,4,5},
{6,7}
n
7→
x1 =x2 , o
x3 =x4 =x5 ,
x6 =x7
7→ S7 .
322
n
x1 =x2 , o
x3 =x4 =x5 ,
x6 =x7
Reflection group counting and q-counting