Transcript Slide 1

Symmetries of Hives,
Generalized Littlewood-Richardson Fillings,
and Invariants of Matrix Pairs over
Valuation Rings.
(preliminary report)
Glenn D. Appleby* and Tamsen Whitehead
Santa Clara University
October 24, 2009
h00
Three Big Ideas:
h10
h20
h30
h40
h50
h21
h31
h41
h51
h11
h22
h32
h42
h52
h33
h43
h53
h44
h54
h55
Hives
Invariants of Matrices over
Valuation Rings
Littlewood-Richardson
Fillings
02
¹1
¹2
¹3
k13
¹ 4 k14 k24 k34
k11
k12
k23
k44
k22
k33
t9
B6 0
6
i nv B
@4 0
0
0
t5
0
0
0
0
t2
0
32
0
1
2
7
6
0 7 6 2t + t
0 5 4 t4 + t3 + t2
t
t3
0
1
2
t +t
t2
0
0
1
t
31
0
C
0 7
7C
0 5A
1
Littlewood-Richardson Fillings
¹1
¹2
¹3
1 k12 1
1 k131
¹ 4 k114 k224 k334
2 k232
2
3 k33 3
1
1
k222
2
1 k11 1
1
1
Entire
shape: ¸
4 k444
º = ( #1’,s, #2’s, #3’s,…)
Given an LR filling of type (¹, º ; ¸), we can replace
• Weakly increasing along rows
thesay
numbers
in boxes
(“content º ”)…
We
the
above
is
a
Littlewood-Richardson
Filling
• Strictly
increasing
in
columns
…with rectangles
whoseoflength
(an “LR” filling)
type (corresponds
¹, º ; ¸). to the
• “Word
Condition”:
multiplicity
of the entry. Let kij denote the number
i’s ini row
j. ¸ # (i+1)’s in rows (i+1) to i+ k + 1
# i’s in of
rows
to i+k
Empty Boxes: ¹.
n
Set: LR(¹, º ; ¸) = f ki j g : f ki j g de¯nes a LR ¯lling of type (¹ ; º ; ¸ )
o
Thus, Littlewood-Richardson fillings can alternately be defined
by inequalities among the {kij}:
1. (LR1) (Sums) For all 1 · j · r , and 1 · i · r ,
Xj
Xr
¹j +
ksj = ¸ j ;
and
ki s = º i :
s= 1
s= i
2. (LR2) (Column St rict ness) For each j , for 2 · j · r and 1 · i · j
we require
¹ j + k1j + ¢¢¢ki j · ¹ ( j ¡
1)
+ k1;( j ¡
1)
+ ¢¢¢+ k( i ¡
1) ;( j ¡ 1) :
3. (LR3) (Word Condit ion) For all 1 · i · r ¡ 1, i · j · r ¡ 1,
j+1
X
Xj
k( i + 1) ;s ·
s= i + 1
ki s :
s= i
Hives: A triangular array of real numbers satisfying
the “rhombus inequalities”.
Sum of Obtuse Entries
¸ Sum of Acute Entries.
h00
h10
h20
h30
(Hives appear in the proof of the
Saturation Conjecture, eigenvalue
problems for Hermitian matrices,
representation theory, etc…)
h40
h50
h21
h31
h41
h51
h11
h22
h32
h42
h52
h33
h43
h53
h44
h54
h55
Hives:
By the Rhombus Inequalities,
differences of consecutive edge
entries yield a decreasing
Partition of Real Numbers:
¹ 1 = h10 ¡ h00
¹
Such a hive is of type (¹ º ; ¸).
¹
We shall let Hive(¹ , º ; ¸) denote
the
¹ set
¸ of¹ all¸hives
¹ ¸of this
¹ ¸type.
¹=
1
(¹ 12; ¹ 2 ; ¹3 3 ; ¹ 44; ¹ 5 )
5
¹
¹
2
3
¹4
1
h00
h10
h20
h30
¸1
¸2
h11
h21
h31
h22
h32
¸
¸3
h33
¸4
Note: the type of a hive ¹ 5 h40 h41 h42 h43 h44
¸5
may be defined with
h50 h51 h52 h53 h54 h55
real-valued “partitions”!
º1
(though hives over the non-negative
integers are often of particular interest.)
º2
º3
º
º4
º5
Invariants of Matrix Pairs
e.g., R = Q[[t]].
Let R denote a discrete valuation ring, with uniformisant t 2 R,
so for all a 2 R, a=utk for some unit u and a non-negative integer k.
We will set kak = k and call this the order of a 2 R.
Let Mr(R) denote the (r £ r) matrices over R, of full rank.
If M 2 Mr(R), then there are invertible P,Q so that:
2
PMQ-1
6
6
= D = 6
4
t
¹
3
1
7
7
7:
5
t¹ 2
..
.
t¹ r
Uniquely determined
by the orbit of M
a diagonal matrix, where 1 ¸ 2 ¸ L r ¸ 0.
We’ll write inv(M) = ¹ = ( ¹1 , …, ¹r )
= The Invariant Partition of M.
Suppose M, N 2 Mr (R) such that
inv(M) = ¹ , inv(N) = º, and inv(MN) = ¸.
We shall let << (M,N) >> denote the orbit of the pair (M,N)
under the action of a triple of invertible matrices
(P,Q,T) 2 GLr(R), acting as:
(P; Q; T) ¢(M ; N ) = (PM Q¡ 1 ; QN T ¡ 1 ):
This action preserves the invariant partition of both
members of the pair, and their product.
Let MR( ¹,º ; ¸) denote the set of orbits of matrix pairs
(M,N) such that inv(M) = ¹ , inv(N) = º, and inv(MN) = ¸.
h00
h10
h20
h30
h40
h50
h21
h31
h41
h51
h11
h22
h32
h42
h52
h33
h43
h53
h44
h54
h55
Hive(¹
, º ; ¸)
Hives
Invariants of Matrix Pairs over
Discrete
Valuation
MR(¹
, º ; ¸) Rings
Littlewood-Richardson
LR(¹
, º ; ¸)
Fillings
02
¹1
¹2
¹3
k13
¹ 4 k14 k24 k34
k11
k12
k23
k44
k22
k33
t9
B6 0
6
i nv B
@4 0
0
0
t5
0
0
0
0
t2
0
32
0
1
2
7
6
0 7 6 2t + t
0 5 4 t4 + t3 + t2
t
t3
0
1
2
t +t
t2
0
0
1
t
31
0
C
0 7
7C
0 5A
1
Pak and Vallejo (2005) determined an injective map:
© : LR(¹ ; º ; ¸ ) ! Hive(¹ ; º ; ¸ ):
where
©(f ki j g) = f hi j g
h00
h10
h20
is given by:
Xq Xp
hpq =
Xq
ki j +
i= 1 j = 1
h30
¹ s:
s= 1
h40
h50
h21
h31
h41
h51
h11
h22
h32
h42
h52
h33
h43
h53
h44
h54
Hence, © ( { kij } ) is a hive over the non-negative integers.
In fact, © is onto the set of all such hives.
©-1 ({ hpq }) = {kij} where kij = (hj,(i-1)+h(j+1)(i)) – (hji+h(j+1)(i-1)), for i < j.
This is just the rhombus difference for right-slanted rhombi.
Thus, ©-1 could be applied to hives of arbitrary type.
h55
To prove the image ©( {kij} ) of some LR
filling is a hive, we must check the
Rhombus Inequalities hold.
For “vertical” rhombi, the inequality reduces to
one of the Column Strictness inequalities in LR2.
h00
h10
For “left-slanted” rhombi, the inequality
reduces to one of the Word Condition
inequalities of LR3.
h20
h30
h11
h21
h31
h22
h32
h33
h40 h41 h42 h43 h44
However, the Rhombus
Inequality for “right-slanted”
h50 h51 h52 h53 h54 h55
rhombi reduces to the condition
kij ¸ 0, but only in the case i < j.
h00
h10
h20
h30
h40
h50
h21
h31
h41
h51
h11
h22
h32
h42
h52
h33
h43
h53
h44
h54
h55
Hivehas(¹been
, º ;known,
¸)
The direction (()
in the case of
R-modules, for some time, but we are interested
in the inverse direction as well, and also more
constructive methods.
©
LR(¹, º ; ¸)
02
¹1
¹2
¹3
k13
¹ 4 k14 k24 k34
k11
k12
k23
k44
MR( ¹,º ; ¸)
k22
k33
t9
B6 0
6
i nv B
@4 0
0
0
t5
0
0
0
0
t2
0
32
0
1
2
7
6
0 7 6 2t + t
0 5 4 t4 + t3 + t2
t
t3
0
1
2
t +t
t2
0
0
1
t
31
0
C
0 7
7C
0 5A
1
LR (¹, º ; ¸) ) MR(¹, º ; ¸)
Theorem (A., 1999):
Let
k ,k , L , k
11
12
The i-th factor:
1r
be a filling in LR(¹, º ; ¸).
k22, L , k2r


krr2
Then set,
2
6
6
6
6
6
4
t
k 11
0
0
..
.
1
t k12
..
.
0
0
..
.
..
0
..
.
.
1
0
t k1r
3
1
6
..
6
.
6
6
1
6
6
6
6
3 2 6
1
6 0 00 ¢¢¢
4
7 66 0 t k 2 2 1 . . .
7 6
7 6 .. . . . . . .
7 ¢66 .
.
.
.
7 6
..
..
5 4 ..
.
.
.
0
¢¢¢ ¢¢¢
0
0
tki i
1
3 . . .2
0
.. 7
6
6
. 7
7
6
7
6
¢¢¢6
0 7
7
6
7
6
4
1 5
t k 2r
1
..
.
0
tki ;r ¡
1
7
7
7
7
7
7
7
7
3
7
¢¢¢ ¢¢¢7 0
5 . 7
.. 1
.. 7
.k i r
7
t
. 7
0 1
.. . . . .
..
.
.
.
.
..
..
. 1
.
0 ¢¢¢ ¢¢¢ 0
..
7
7
7
0 5
tkr r
=N
With N defined as above, and
2
t
¹
1
6
6 0
M = 6
6 .
4 ..
0
0
t¹ 2
..
.
¢¢¢
¢¢¢
..
.
..
.
0
3
0
.. 7
. 7
7;
7
0 5
The fact that
{kij}2 LR( ¹ , º ; ¸ )
implies the matrices
have the correct
invariants.
t¹ r
then,
inv(M) =  = (1, 2, L , r),
inv(N) =  = ( 1, 2, L , r ),
And inv(MN) =  = (1, 2, L , r).
Note: Matrix Realizations of LR fillings of conjugate type
were first obtained by Sa and Azenhas in 1990.
This Construction gives us an injective map:
ª : LR(¹ ; º ; ¸ ) ! M R (¹ ; º ; ¸ )
Recent work (A., Whitehead, 2009) has led to a left inverse
to the above, that is, a surjective map:
ªe : M R (¹ ; º ; ¸ ) ! LR(¹ ; º ; ¸ )
such that ªe ± ª = I d:
However, there are inequivalent orbits in MR(¹, º ; ¸)
giving rise to the same filling in LR(¹, º ; ¸).
The associated LR filling is only a discrete invariant of
the orbit.
Our met hod is t o ¯nd in t he orbit < < (M ; N ) > > a dist inguished
pair
(D ¹ ; L D ºb ) 2 < < (M ; N ) > > ;
where
2
6
6
D¹ = 6
6
4
t¹ 1
0
0
..
.
t¹ 2
..
.
¢¢¢
0
¢¢¢
..
.
..
.
0
3
2 ºr
0
t
6
.. 7
7
6 0
. 7
6
7 ; D ºb = 6 .
4 ..
0 5
t¹ r
0
0
tº r ¡ 1
..
.
¢¢¢
3
0
.. 7
. 7
7;
7
0 5
¢¢¢
..
.
..
.
0
tº 1
and L is a special, invert ible, lower-t riangular mat rix
(called ¹ -b
º -generic) sat isfying a class of inequalit ies
among t he orders of t he det erminant s of it s minors.
Theorem (A., Whitehead, 2009): Given a ¹ -b
º -generic
matrix L, in the orbit of a pair (M,N), then for i < j:
° ³
^
°
° L (j ¡ i ) ; : : :
°
1
:::
; (j ¡ 1)
(r ¡ i )
^
´ °°
°¡
°
Denotes the order of the minor of L
with the given rows and columns.
° ³
^
°
° L (j ¡ i + 1) ; : : :
°
1
:::
´ °°
; (j )
°
(r ¡ i ) °
^
= k1j + k2j + ¢¢¢+ ki j ;
where {kij} 2 LR( ¹, º ; ¸), and (j-i)^ denotes
removing the indicated row from the minor.
The filling is an invariant of the orbit.
A variation of the above formula can be used to compute
the corresponding hive entry in Hive(¹ , º ; ¸).
Now, we can connect
Hives to LR fillings,
and LR fillings to
Matrix Pair Invariants.
…Provided everything
is defined over the
non-negative integers.
h00
h10
h20
h30
h40
h50
h21
h31
h41
h51
h11
h22
h32
h42
h52
h33
h43
h53
h44
h54
h55
Hive (¹, º ; ¸)
LR(¹, º ; ¸)
02
¹1
¹2
¹3
k13
¹ 4 k14 k24 k34
k11
k12
k23
k44
MR( ¹,º ; ¸)
k22
k33
t9
B6 0
6
i nv B
@4 0
0
0
t5
0
0
0
0
t2
0
32
0
1
2
7
6
0 7 6 2t + t
0 5 4 t4 + t3 + t2
t
t3
0
1
2
t +t
t2
0
0
1
t
31
0
C
0 7
7C
0 5A
1
Hives are de¯ned over R, so, theimage©¡ 1(Hive(¹ ; º ; ¸ ))
can be computed for hives of arbitrary, real valued type.
If ©¡ 1(f hpqg) = f ki j g, by t he Rhombus Inequalit ies, the part s ki j of the\ ¯lling" de¯ned
by t his inverse will sat isfy:
Call these
“interior parts”
ki j ¸ 0; whenever i < j .
Call these
“edge parts”
However, it is possible that a part ki i might
be negative, for some i ...
...and,all part s ki j as well as the parts of the
partitions ¹ , º , and ¸ will be r eal valued.
Are these “fillings” ©-1 (Hive(¹, º ; ¸)) realized as an
invariants of an actual objects of study?
Our original matrix results were over discrete
valuation rings, leading to LR fillings over the
non-negative integers.
We have extended these results to a class of rings
with a real-valued valuation.
There are many such examples. For one, let F
denote a field of formal power series:
X1
a=
ci t ®i 2 F
i= 0
with real-valued exponents ®i. We’ll suppose the
set of exponents for any a 2 F has no limit points, so
multiplication in F is well-defined.
With each element a =
X1
ci t ®i 2 F we may define the order of a as:
i= 0
kak = ®0 2 R:
We shall define R µ F as the valuation ring:
R = f a 2 F : kak ¸ 0g;
Whose units are the set: R£ = f a 2 R : kak = 0g:
T heor em 1 Let M 2 M r (F ). T hen there exist P; Q 2
GL r (R) such that
P M Q¡
Over R,
Not F
1
= di ag(t ¹ 1 ; t ¹ 2 ; : : : ; t ¹ r );
where ¹ 1 ¸ ¹ 2 ¸ ¢¢¢ ¸ ¹ r . The r eal-valued exponents
i nv(M ) = ¹ = (¹ 1 ; ¹ 2 ; : : : ; ¹ r ) are uniquely determined by
M and are an invariant of the orbit under the action of
matrix equivalence.
We now work with full-rank matrix pairs (M,N) 2 Mr( F ),
And, as before, will study the orbits of the action
(P; Q; T) ¢(M ; N ) = (PM Q¡ 1 ; QN T ¡ 1 ):
Where pairs (M,N) 2 Mr( F ), but (P,Q,T) 2 GLr(R).
In this sense, we can form the set of orbits,
denoted by: M F(¹,º ;¸), where ¹, º, and ¸ are now
partitions of real numbers.
Essentially, everything goes over to the R-valued case.
All our constructions extend to this setting, including the
determinantal formulas for fillings.
That is, given an orbit of a matrix pair (M,N) in M F (¹, º ; ¸) , we can
find GLr (R) invariants that allow us to define real numbers {kij} that
formally satisfy:
1. (LR1) (R-Sums) For all 1 · j · r , and 1 · i · r ,
Xj
Xr
¹j +
ksj = ¸ j ;
and
ki s = º i :
s= 1
s= i
2. (LR2) (R-Column St rict ness) For each j , for 2 · j · r and 1 · i · j
we require
¹ j + k1j + ¢¢¢ki j · ¹ ( j ¡
1)
+ k1;( j ¡
1)
+ ¢¢¢+ k( i ¡
1) ;( j ¡ 1) :
3. (LR3) (R-Word Condit ion) For all 1 · i · r ¡ 1, i · j · r ¡ 1,
j+1
X
Xj
k( i + 1) ;s ·
s= i + 1
ki s :
s= i
However, using these formulas, we find there is an additional
property our “ LR” fillings satisfy:
4. (LR4) (R-Non-negat ivity) For all int erior part s,
where i < j , we have ki j ¸ 0. There is no condit ion on t he edge part s ki i except ki i 2 R.
That is, fillings {kij} from matrix pairs over F
yield the same fillings
as
©-1(Hive(¹, º; ¸)) under the Pak-Vallejo map:
³
´
³
´
ªe M F (¹ ; º ; ¸ ) = ©¡ 1 Hive(¹ ; º ; ¸ )
This appears to be the “right” generalization to
real-valued LR fillings.
So, we can determine a “filling” {kij}.
Can we draw the diagram?
Begin by drawingInterior
We measure
parts (always
to the right
non-negative)
for positive
areparts,
added,
the “base”, ¹. moving
and toto
thethe
leftright:
for negative parts.
¹1
¹¹ 11
¹2
¹¹ 22
¹¹ 33
¹¹ 44
k14
14
¹¹ 4
4
¹¹ 3
3
k 24
k13
¸1
¸2
k11
k12k12 k22
k23k23 k33
k34k34 k44
Edge parts, when negative,
move the end of the row to
the left.
Insert an “origin”,
We shall call the
measured horizontally
¸3
above a diagram of a real-valued
Littlewood-Richardson filling, or
The partition ¸ is measured froman
the“ origin
to
LR“ filling.
the end of the row (here, positive, though a
negative row is possible).
¸4
h00
So now, over R, we
can connect:
h10
h20
h30
h40
h50
h21
h31
h41
h51
h11
h22
h32
h42
h52
h33
h43
h53
h44
h54
h55
Hive(¹, º ; ¸)
M F (¹ ; º ; ¸ )
LR(¹ ; º ; ¸ )
¹1
¹2
k14
¹4
¹3
k24
k13
k12
k23
k34
k33
k44
k11
k22
02
t9
B6 0
6
i nv B
@4 0
0
0
t5
0
0
0
0
t2
0
32
0
1
2
7
6
0 7 6 2t + t
0 5 4 t4 + t3 + t2
t
t3
0
1
2
t +t
t2
0
0
1
t
31
0
C
0 7
7C
0 5A
1
I. From Hives, to Matrices and LR Fillings
h00
h10
h20
h30
h40
h50
h21
h31
h41
h51
h11
h22
h32
h42
h52
h33
h43
h53
h44
h54
h55
hh0000º ; ¸)
Hive(¹,
hh10
h
10 h1111
hh2020 hh2121 hh2222
hh3030 hh31
31 hh3232 hh
3333
hh4040 hh4141 h4242 hh4343 h44h44
¹
¸
h50h50 hh5151 h52
52 hh5353 hh
5454 h55h55
º
Note that, if inv(M) = ¹, then i nv(M ¡ 1 ) = ¹e:
We will depict a hive of this type by:
h00
50
º¹
º
h10
51 h11
40
h20
52 h21
41 h22
30
h30
53 h31
42 h32
31 h33
20
= (¡ ¹ r ; ¡ ¹ r ¡ 1 ; : : : ; ¡ ¹ 1 )
¹¹¸e¹e
h40
54 h41
43 h42
32 h43
21 h44
10
h50
55 h51
44 h52
33 h53
22 h54
11 h55
00
¸¸ºe
e
¸
= (¡ ¸ r ; ¡ ¸ r ¡ 1 ; : : : ; ¡ ¸ 1 )
Inv(M)
Inv(¤)
¹
¹º
¸
º
Inv(N)
MN= ¤
Set ¤ = MN, so that
Vert. Re° ection
¹¸
¹¸
2
3 clock-wise
1
3 clock-wise
¹¸e
¹¸e
¸ºe
¹º
º
¸e
N¤-1 = M-1
¤-1 M = N-1
Vert. Re° ection
+ 13 clock-wise
Vert. Re° ection
+ 23 clock-wise
¹¸e
¹¸ºe
ºe
¹ºe
¤ N-1 = M
N-1 M-1 = ¤-1
º
¸e
¹ºe
¹¸e
M-1 ¤ = N
So, with six different orientations of
hives… come six different, but
related LR Diagrams.
¹
¸
The fillings in ºthese six
¹e
º
¹e
diagrams are linearly
¸
related
by the S3
¸e
symmetries of the (linearly
ºe
¸e related)
entries of the ºe
¹
¹e
corresponding
hives.
¹
¸
ºe
º
¸e
II. From LR Fillings to Matrices and Hives
In classical LR fillings, proving c¹ º¸ = cº, ¹¸ is established
by constructing a bijection
LR(¹, º ; ¸)
,
LR(º, ¹ ; ¸)
as found in Kerber and James, and generalized by the row-switching
algorithms of Benkhart, Sottile, and Stroomer (1996).
What does this relationship among LR fillings suggest
about matrix invariants and hives?
Recall we calculate an LR filling from matrices by using the order
of a minor of the L:
° ³
^
°
° L (j ¡ i ) ; : : :
°
1
:::
; (j ¡ 1)
(r ¡ i )
^
´ °°
°
°
2
{kij} 2 LR (¹ , º ; ¸)
L=
1
6
6 a21
6
6 ..
6 .
6
6 ..
6 .
6
6 .
6 ..
6
6 .
4 ..
ar 1
Omitted rows
3
0 ::: ::: ::: :::
.. ..
.
.
.. .. ..
.
.
.
.. .. ..
.
.
.
..
..
.
.
..
..
.
.
:::
ar ;r ¡ 2
0
.. 7
. 77
.. 7
. 77
.. 7
. 77
.. 77
.7
7
05
1
{mij} 2 LR (º ,¹ ; ¸)
That is, we may use a ¹ -b
º -generic matrix L, to get two LR fillings,
the first in LR( ¹ , º ; ¸) as before, and a second in LR( º , ¹ ; ¸):
(M,N)
“content ¹ ”
{ mij } 2 LR( º , ¹ ; ¸ )
(“left filling”)
“content º ”
{ kij } 2 LR( ¹ , º ; ¸ )
(“right filling”)
Theorem (A., Whitehead, 2009): Given any matrix pair (M,N)2 M r (F ) , and
any right filling associated to the orbit of the pair, there is a unique left filling
also associated to the pair that is independent of the matrix realization of the
filling. Further, if ¹, º, and ¸, along with the filling, are realized by nonnegative integers, this algebraic bijection of fillings is the same as the
combinatorial bijection described by Kerber and James for classical LR fillings.
Theorem (A., Whitehead, 2009) : In terms of hives, this
bijection on LR fillings and matrix pairs:
yields a new involution on hives, (which we can
describe independently of the filling) taking
each hive of the form:
¹
¸
º
into one of the
form:
º
¸
¹
Note that, in particular, this new hive is not obtainable
by any of the S3 symmetries found earlier.
III. From Matrix Pairs to LR Fillings and Hives
³
´
¡
4
¡
4
¡
4
(DD¹¹ ; L ¢D
¢Dºbºb ¢di
) =ag(t ; t ; : : : ; t ) =
00 22 8 8
3 32 2
t t 0 0 0 00 0
1 1
0
00
B66 00 t 5t 5 0 0 0 70 76 62 t 2 +2 tt2 + t 1
B
01
B66
7
6
B
7
6
;
;
4
@
@44 00 0 0 t 2 t 20 50 54 t 4
+ t t43 ++ t 32 + t 2t 2+ t t 2 1+
00 0 0 0 0 t t
t3 t3
t2
tt 2
“Base” º
(Left) LR Filling:
0
0
t0
1
32
0
706
76
514
t
2¡
32
4
2
t0
t0
6 3¡ 4
007
7 6t 0
005 4 0
0
10
3311
0 0 0 00
7C
t 3 0 0 00 7
7
7C
6¡ 4 6
5
00 5A
0t t
0 0 0 t 11t ¡114
(Right) LR Filling:
Content º
4
The Left Filling is unchanged,
though the origin is shifted.
For scalar shifts of the left member, these
phenomena are reversed.
Scalar shift on the right only
shifts the edge parts of the
Right Filling – Interior parts are
invariant under scalar shifts.
There’s a nice description of the
scalar shift of a hive, too!
Our interest is not so much in counting fillings of a given type, but to
understand the interconnections between seemingly different fillings, and
the objects that relate them.
(M,N)
³
´
D ¹ ; L D ºb
LR Filling of ¸ / º with content ¹
…where both sides are related
by the “left-right” LR bijections.
Then, by the hive symmetries,
we obtain the related fillings:
LR Filling of ¸ / ¹ with content º
(“left filling”)
¹e
¸e
º
º
¸
ºe
º
¸
¸e
ºe
¹
¸
¹
¸e
¸e
¹e
¸
ºe
ºe
¹
º
¹e
¹e
¸e
¹e
º
¹
(“right filling”)
¸e
ºe
¹
¸
¹
¸
ººe
¹e
(M,N)
Etc….
³
³
And, for each family of fillings,
we obtain a new family (with
the same interior parts) by
various scalar shifts:
´
D ¹ ; L D ºb
´
D (· )D ¹ ; L D ºb D (!
¹
¸
º
º
¸
¸e
¸e
º
¹e
¹e
ºe
¸e
ºe
¹
¹e
¹
¸
ºe
º
¹
¸
¹
´
º
¸
D (° )D ¹¹e ; L D ºb D (´
º
¹
º
³
¹e
¸e
¹
ºe
º
¸e
¹e
D (®)D ¹ ; L D ºb D (¯
¹
º
º
ºe
ºe
¹
¹e
¹
¸
¸
¸e
ºe
¸e
ºe
¹
ºe
¹e
º
¹
¸
¸e
ºe
¸e
ºe
¹
¸
¸e
ºe
¹e
¹e
¸e
¸e
º
¹e
¹e
¸
º
¸
¹
´
º
¸e
¸e
¹e
¸e
¸
³
¸
¹eº
ºe
¸
¹e
¹
¸
ºe
ºe
These results imply, among other things, that huge families
of scalar-shifted, “left-right”-related, or hive-symmetryrelated triples of partitions have equal LR coefficients, in
that they all share the same “combinatorial core”...
…and there’s more...
…but time’s up! Thank you!
¹
¸
º
º
¸e
º
¹e
¹e
¸
¸e
ºe
¸e
ºe
¹
¹e
¹
¸
ºe
What is the Hive corresponding to the scalar shift
³
´
D ¹ ; L D ºb
³
´
D ¹ ; L (D ºb ¢D (¡ 4) ?
!
7
¸ + (¡ 4)¸ = (10;
(14; 6;
10;4;8;2)6)
00
8
¹ = (8; 5; 2; 1)
2
2
1
15
16
º = ( 11;
º + (¡ 4)
7;
8
5
13
14
10
14¡
10
14 4
21¡
17
21 4
25¡
21
25 4
27¡
23
27 4
6;
2;
¡ 1;
3;
10
6
24¡ 8
16
24
30¡
22
30 8
33¡
25
33 8
1
4
8
1
32¡ 12
20
32
36¡
24
36 12
¡ 22 )
6
2
22
3838¡ 16
A shift of the left member corresponds to a adding a “staircase”
down the hive :
³
´
D ¹ ; L D ºb
³
!
7
´
D (¡ 7) ¢D ¹ ; L (D ºb )
¸ + (¡ 7)¸ == (7;
1; ¡8;1)6)
(14;3;10;
¹ + (¡ 7)
8
¹ == (1;
(8;¡5;2;2;¡ 1)
5; ¡ 6)
00
8¡
8¡
8¡
81777
5
14
714¡
14¡
14¡
77 ³
14¡7710
14
Pleasant Exercise:
Calculate the hive and
diagram for:
D (¡ 7) ¢D ¹ ; L (D ºb D (+ 7))
13¡
¡14
14
1
13
2 13¡
¡216
1 15¡1515
16¡
¡ 28
12
16
1616
º = (
25¡
25
25421
27¡
27
¡281
27
27
11;
21¡
7 14
21
930
30¡
30
32¡
30¡ 21
21 113232
32¡ 21
216
33¡
5 28
33
33
6;
1024
24¡
24¡
14
24¡14
14
8
3;
83636
36¡
36 28 10383838¡
38 28
2
)
´
Given the Hive:
º
¹e
Where: ¹ = (8; 5; 2; 1),
º = (11; 6; 3; 2), and
¸ = (14; 10; 8; 6).
¸e
We first depict the (non-negative)
partition º :
Which results in the shape ¹e .
We then extend the diagram by the
non-negative interior parts kij, of the
filling ¸e where i < j…
…followed by the (negative) edge
parts kii:
¸e
ºe
¹
¹
¸
ºe
¸e
ºe
¹e