Grothendieck expansions of symmetricpolynomials
Jason Bandlow (joint work with Jennifer Morse)
University of Pennsylvania
August 3rd, 2010 – FPSACSan Francisco State University
Outline
Symmetric functions
Tableaux–Schur expansions
Grothendieck functions and their dual basis
Main theorem
Examples
Sketch of proof
The monomial basis
The monomial symmetric functions are indexed by partitions
λ = (λ1, λ2, . . . , λk) λi ≥ λi+1
mλ =∑α
xα
α a rearrangement of the parts of λ and infinitely many 0’s
Example
m2,1 =(x21x2 + x1x2
2 ) + (x1x23 + x2
1x3) + . . .
+ (x22x3 + x2x2
3 ) + . . .
The complete homogeneous basis
The (complete) homogeneous symmetric functions are defined by
hi =∑λ`i
mλ
hλ = hλ1hλ2 . . . hλk
Example
h3 = m3 + m2,1 + m1,1,1
h4,2,1 = h4h2h1
The Hall inner product
Defined by
〈hλ,mµ〉 =
{1 if λ = µ
0 otherwise
Proposition
If {fλ}, {f ∗λ } and {gλ}, {g∗λ} are two pairs of dual bases with
fλ =∑µ
Mλ,µgµ
then
g∗µ =∑λ
Mλ,µf ∗λ
The Hall inner product
Defined by
〈hλ,mµ〉 =
{1 if λ = µ
0 otherwise
Proposition
If {fλ}, {f ∗λ } and {gλ}, {g∗λ} are two pairs of dual bases with
fλ =∑µ
Mλ,µgµ
then
g∗µ =∑λ
Mλ,µf ∗λ
Semistandard Young tableaux
A left-and-bottom justified, partition-shaped array of numbers,weakly increasing across rows and strictly increasing up columns.
Example
7 75 6 6 83 3 4 71 1 2 2 2
Shape: (5, 4, 4, 2)Evaluation: (2, 3, 2, 1, 1, 2, 3, 1)Reading word: 775668334711222
Semistandard Young tableaux
A left-and-bottom justified, partition-shaped array of numbers,weakly increasing across rows and strictly increasing up columns.
Example
7 75 6 6 83 3 4 71 1 2 2 2
Shape: (5, 4, 4, 2)Evaluation: (2, 3, 2, 1, 1, 2, 3, 1)Reading word: 775668334711222
Knuth equivalence
An equivalence relation on words generated by
yxz ≡ yzx if x < y ≤ z
xzy ≡ zxy if x ≤ y < z
Key Fact
Every word is Knuth equivalent to the reading word of exactly onetableau.
Example
rw
(3 41 2 3
)= 34123 ≡ 31423 ≡ 31243 ≡ 13243 ≡ 13423
The Schur basis
DefinitionThe Schur functions are given by
sλ =∑
T∈SSYT (λ)
xev(T )
Example
s2,1 = x21x2+ x1x2
2+ 2x1x2x3+ · · ·21 1
21 2
31 2
21 3 · · ·
Fact: Schur functions are a self-dual basis of the symmetricfunctions.
The Schur basis
DefinitionThe Schur functions are given by
sλ =∑
T∈SSYT (λ)
xev(T )
Example
s2,1 = x21x2+ x1x2
2+ 2x1x2x3+ · · ·21 1
21 2
31 2
21 3 · · ·
Fact: Schur functions are a self-dual basis of the symmetricfunctions.
The Schur basis
DefinitionThe Schur functions are given by
sλ =∑
T∈SSYT (λ)
xev(T )
Example
s2,1 = x21x2+ x1x2
2+ 2x1x2x3+ · · ·21 1
21 2
31 2
21 3 · · ·
Fact: Schur functions are a self-dual basis of the symmetricfunctions.
The Schur basis
Using the fact that Schur functions are symmetric, we can rewritethe definition as
sλ =∑µ
Kλ,µmµ
where Kλ,µ is the number of semistandard tableaux of shape λ andevaluation µ.
Using the proposition about dual bases, we get
hµ =∑λ
Kλ,µsλ
which can be rewritten as
hµ =∑T∈Tµ
ssh(T )
where Tµ is the set of all semistandard tableaux of evaluation µ.
The Schur basis
Using the fact that Schur functions are symmetric, we can rewritethe definition as
sλ =∑µ
Kλ,µmµ
where Kλ,µ is the number of semistandard tableaux of shape λ andevaluation µ.Using the proposition about dual bases, we get
hµ =∑λ
Kλ,µsλ
which can be rewritten as
hµ =∑T∈Tµ
ssh(T )
where Tµ is the set of all semistandard tableaux of evaluation µ.
The Schur basis
Using the fact that Schur functions are symmetric, we can rewritethe definition as
sλ =∑µ
Kλ,µmµ
where Kλ,µ is the number of semistandard tableaux of shape λ andevaluation µ.Using the proposition about dual bases, we get
hµ =∑λ
Kλ,µsλ
which can be rewritten as
hµ =∑T∈Tµ
ssh(T )
where Tµ is the set of all semistandard tableaux of evaluation µ.
Tableaux–Schur expansions
Elements of a family {fα} of symmetric functions havetableaux–Schur expansions if there exist sets Tα of semistandardtableaux and weight functions wtα such that
fα =∑T∈Tα
wtα(T )ssh(T )
Goal: find appropriate sets and modifications of wt so that
fα =∑S∈Sα
wtα(S)Gsh(S)
fα =∑R∈Rα
wtα(R)gsh(R)
where G and g are, respectively, the Grothendieck anddual-Grothendieck functions.
Tableaux–Schur expansions
Elements of a family {fα} of symmetric functions havetableaux–Schur expansions if there exist sets Tα of semistandardtableaux and weight functions wtα such that
fα =∑T∈Tα
wtα(T )ssh(T )
Goal: find appropriate sets and modifications of wt so that
fα =∑S∈Sα
wtα(S)Gsh(S)
fα =∑R∈Rα
wtα(R)gsh(R)
where G and g are, respectively, the Grothendieck anddual-Grothendieck functions.
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
Tableaux–Schur expansions
Examples
I hµ =∑
T∈Tµ ssh(T )
I Hµ[X ; t] =∑
T∈Tµ tch(T )ssh(T )
I sλsµ =∑
T∈Tλ,µ ssh(T )
I Fσ =∑
T∈Tσ ssh(T )
I A(k)µ [X ; t] =
∑T∈Tk,µ
tch(T )ssh(T )
I Hµ[X ; 1, t] =∑
T∈T(1n)tchµ(T )ssh(T )
I Hµ[X ; q, t]?=∑
T∈T(1n)qaµ(T )tbµ(T )ssh(T )
Grothendieck polynomials
I Introduced by Lascoux-Schutzenberger (1982)
I Represent K -theory classes of structure sheaves of Schubertvarieties in GLn
I Given by elements of Z[[x1, . . . , xn]]
I Analogous to Schubert polynomials
I Sign-alternating by degree, equal to Schubert polynomial inbottom degree
Stable Grassmannian Grothendieck functions
Stable:
I Due to Fomin-Kirillov
I Limit as number of variables →∞
Grassmannian:
I Indexed by partitions
I Power series of symmetric functions
I Sign-alternating by degree in the Schur basis
I Analog of Schur functions
Stable Grassmannian Grothendieck functions
Stable:
I Due to Fomin-Kirillov
I Limit as number of variables →∞Grassmannian:
I Indexed by partitions
I Power series of symmetric functions
I Sign-alternating by degree in the Schur basis
I Analog of Schur functions
Grassmannian Grothendieck functions
Theorem (Buch)
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
where SVT (λ) is the set of set-valued tableaux of shape λ.
Example
G1 = s1 −s1,1 +s1,1,1 . . .
1 12 123 · · ·
23 234 · · ·...
...
Grassmannian Grothendieck functions
Theorem (Buch)
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
where SVT (λ) is the set of set-valued tableaux of shape λ.
Example
G1 = s1 −s1,1 +s1,1,1 . . .
1 12 123 · · ·
23 234 · · ·...
...
Set-valued tableaux
Example
234 45
1 12 3
shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1)
ε(S) = (−1)|ev(S)|+|sh(S)| = 1
From
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
we seeGλ = sλ ± higher degree terms
Set-valued tableaux
Example
234 45
1 12 3
shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1)
ε(S) = (−1)|ev(S)|+|sh(S)| = 1
From
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
we seeGλ = sλ ± higher degree terms
Set-valued tableaux
Example
234 45
1 12 3
shape(S) = (3, 2) evaluation(S) = (2, 2, 2, 2, 1)
ε(S) = (−1)|ev(S)|+|sh(S)| = 1
From
Gλ =∑
S∈SVT (λ)
ε(S)xev(S)
we seeGλ = sλ ± higher degree terms
Set-valued tableaux
We introduce the reading word of a set-valued tableau, defined by:
I Proceed row by row, from top to bottom
I In each row, first ignore the smallest element of each cell.Then read the remaining elements from right to left, and fromlargest to smallest within each cell.
I Read the smallest elements of each cell from left to right.
Example
234 45
1 12 3
has reading word: 543242113.
Dual Grothendieck functionsWe denote the dual basis to the Grothendieck by g
Theorem (Lam, Pylyavskyy)
gλ =∑
R∈RPP(λ)
xev(R)
where RPP(λ) is the set of reverse plane partitions of shape λ.
Example
g2,1 = s2,1 +s2
21 1
11 1
21 3
11 2
......
Dual Grothendieck functionsWe denote the dual basis to the Grothendieck by g
Theorem (Lam, Pylyavskyy)
gλ =∑
R∈RPP(λ)
xev(R)
where RPP(λ) is the set of reverse plane partitions of shape λ.
Example
g2,1 = s2,1 +s2
21 1
11 1
21 3
11 2
......
Reverse plane partitions
Example
2 32 2 41 2 21 1 1 1
shape(R) = (4, 3, 3, 2) evaluation(R) = (4, 3, 1, 1)
From
gλ =∑
R∈RPP(λ)
xev(R)
we seegλ = sλ ± lower degree terms
Reverse plane partitions
Example
2 32 2 41 2 21 1 1 1
shape(R) = (4, 3, 3, 2) evaluation(R) = (4, 3, 1, 1)
From
gλ =∑
R∈RPP(λ)
xev(R)
we seegλ = sλ ± lower degree terms
Dual Grothendieck functions
We define the reading word of a reverse plane partition to be asubsequence of the usual reading word, where we only take thebottommost occurence of every letter in each column.
Example
2 32 2 41 2 21 1 1 1
has reading word 324221111.
Generalizing tableaux–Schur expansions
Given any set Tα of semistandard tableaux, we definecorresponding sets Sα (respectively Rα) of set-valued tableaux(reverse plane partitions) defined by
S ∈ Sα ⇐⇒ rw(S) ≡ rw(T ) for some T ∈ Tα
R ∈ Rα ⇐⇒ rw(R) ≡ rw(T ) for some T ∈ Tα
Given a statistic wtα on Tα, we can extend it to reading words bywtα(w) = wtα(T ) if w ≡ rw(T ).
Generalizing tableaux–Schur expansions
Given any set Tα of semistandard tableaux, we definecorresponding sets Sα (respectively Rα) of set-valued tableaux(reverse plane partitions) defined by
S ∈ Sα ⇐⇒ rw(S) ≡ rw(T ) for some T ∈ Tα
R ∈ Rα ⇐⇒ rw(R) ≡ rw(T ) for some T ∈ Tα
Given a statistic wtα on Tα, we can extend it to reading words bywtα(w) = wtα(T ) if w ≡ rw(T ).
Examples
If T =
43 41 2 ∈ Tα, then
43 41 2
431 24
34 41 2
4 41 23
4
1 234
43412 43412 43412 44312 44312
are in Sα
and
43 41 2
4 41 31 2
4 43 41 2
44 41 31 2 · · ·
43412 44312 43412 44312
are in Rα. All of these will have the same weight.
Examples
If T =
43 41 2 ∈ Tα, then
43 41 2
431 24
34 41 2
4 41 23
4
1 234
43412 43412 43412 44312 44312
are in Sα and
43 41 2
4 41 31 2
4 43 41 2
44 41 31 2 · · ·
43412 44312 43412 44312
are in Rα. All of these will have the same weight.
Statement of main theorem
Theorem (B-Morse)
Let fα be a family of symmetric functions with
fα =∑T∈Tα
wtα(T )ssh(T ).
Then
fα =∑S∈Sα
ε(S)wtα(S)gsh(S)
and
fα =∑R∈Rα
wtα(R)Gsh(R).
Complete homogeneous functions
We have a tableaux-Schur expansion of the homogeneous functionsby
hµ =∑T∈Tµ
ssh(T )
where Tµ is the set of all tableaux of evaluation µ. Thus we have
hµ =∑S∈Sµ
ε(S)gsh(S) =∑R∈Rµ
Gsh(R)
where Sµ is the set of all set-valued tableaux of evaluation µ andRµ is the set of all reverse plane partitions of evaluation µ.
Example
h3,2 = s3,2 +s4,1 +s5
2 21 1 1
21 1 1 2 1 1 1 2 2
= g3,2 +g4,1 +g5
−g3,1 −g4
21 1 12 1 1 12 2
= G3,2 +G4,1 +G5
+2G3,2,1 +2G4,1,1 + · · ·22 21 1 1
21 21 1 1
221 1 1 2
211 1 1 2
Example
h3,2 = s3,2 +s4,1 +s5
2 21 1 1
21 1 1 2 1 1 1 2 2
= g3,2 +g4,1 +g5
−g3,1 −g4
21 1 12 1 1 12 2
= G3,2 +G4,1 +G5
+2G3,2,1 +2G4,1,1 + · · ·22 21 1 1
21 21 1 1
221 1 1 2
211 1 1 2
Example
h3,2 = s3,2 +s4,1 +s5
2 21 1 1
21 1 1 2 1 1 1 2 2
= g3,2 +g4,1 +g5
−g3,1 −g4
21 1 12 1 1 12 2
= G3,2 +G4,1 +G5
+2G3,2,1 +2G4,1,1 + · · ·22 21 1 1
21 21 1 1
221 1 1 2
211 1 1 2
Hall-Littlewood functions
The Hall-Littlewood functions form a basis for the symmetricfunctions over the field Q(t). Interpretations as
I Deformation of Weyl character formula
I Graded Sn-character of certain cohomology rings
I Representation theory of groups of matrices over finite fields
among others.
We will use the version typically denoted by Hµ[X ; t] or Q ′µ(x ; t) inthe literature.
Example
H1,1,1[X ; t] = s1,1,1 + (t2 + t)s2,1 + t3s3
Hall-Littlewood functions
The Hall-Littlewood functions form a basis for the symmetricfunctions over the field Q(t). Interpretations as
I Deformation of Weyl character formula
I Graded Sn-character of certain cohomology rings
I Representation theory of groups of matrices over finite fields
among others.
We will use the version typically denoted by Hµ[X ; t] or Q ′µ(x ; t) inthe literature.
Example
H1,1,1[X ; t] = s1,1,1 + (t2 + t)s2,1 + t3s3
Hall-Littlewood functions
Theorem (Lascoux-Schutzenberger)
Hµ[X ; t] =∑T∈Tµ
tch(T )ssh(T )
where charge is a non-negative integer statistic defined on wordsand constant on Knuth equivalence classes.
The charge statistic
Example
3 42 2 31 1 1 2
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0 0
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0 0 0
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 0
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 0 0
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 1 0 0
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 1 0 0 0
3 4 2 2 3 1 1 1 2
The charge statistic
Example
3 42 2 31 1 1 2
0 1 0 0 1 0 0 0 1
3 4 2 2 3 1 1 1 2
charge(T ) = 3
Hall-Littlewood functions
We have
Hµ[X ; t] =∑T∈Tµ
tch(T )ssh(T )
where Tµ is again the set of all tableaux of evaluation µ. Hence
Hµ[X ; t] =∑S∈Sµ
ε(S)tch(S)gsh(S)
=∑R∈Rµ
tch(R)Gsh(R)
where Sµ (resp. Rµ) is the set of set-valued tableaux (resp. reverseplane partitions) of evaluation µ and the charge of S or R is thecharge of the reading word.
Example
H1,1,1[X ; t] = s1,1,1 + (t + t2)s2,1 + t3s3
=g1,1,1 + (t + t2)g2,1 + t3g3
− 2g1,1 − (t + t2)g2 + g1
321
21 3
31 2 1 2 3
231
312 12 3 1 23 123
Example
H1,1,1[X ; t] = s1,1,1 + (t + t2)s2,1 + t3s3
=g1,1,1 + (t + t2)g2,1 + t3g3
− 2g1,1 − (t + t2)g2 + g1
321
21 3
31 2 1 2 3
231
312 12 3 1 23 123
Littlewood-Richardson tableaux
One version of the Littlewood-Richardson rule states that
sλsµ =∑
T∈Tλ,µ
ssh(T )
where Tλ,µ is the set of tableaux:
I with evaluation (λ1, · · · , λ`(λ), µ1, · · · , µ`(µ))I which have the reverse lattice property with respect to the
letters 1, · · · , `(λ), and
I which have the reverse lattice property with respect to theletters `(λ) + 1, · · · , `(λ) + `(µ)
Littlewood-Richardson tableaux
Example
T(2,1),(2,1) is
2 41 1 3 3
421 1 3 3
2 3 41 1 3
42 31 1 3
32 41 1 3
4321 1 3
3 42 31 1
432 31 1
Littlewood-Richardson tableaux
Corollary
sλsµ =∑
S∈Sλ,µ
ε(S)gsh(S) =∑
R∈Rλ,µ
Gsh(R)
where Sλ,µ (resp. Rλ,µ) is the set of all set-valued tableaux (resp.reverse plane partitions)
I with evaluation (λ1, · · · , λ`(λ), µ1, · · · , µ`(µ))I which have the reverse lattice property with respect to the
letters 1, · · · , `(λ), and
I which have the reverse lattice property with respect to theletters `(λ) + 1, · · · , `(λ) + `(µ)
Grothendieck and Schur functions
Corollary
sλ =∑
S∈Sλ,∅
ε(S)gsh(S)
sλ =∑
R∈Rλ,∅
Gsh(S)
Duality gives expansions of G and g into Schur functions.A different form of these expansions was given by Lenart, in termsof combinatorial objects now called elegant fillings.
Sketch of proof
Given
fα =∑T∈Tα
wt(T )ssh(T )
=∑T∈Tα
wt(T )∑
T ′∈SSYT (sh(T ))
xev(T′)
we want to show
fα =∑S∈Sα
ε(S)wt(S)gsh(S)
=∑S∈Sα
ε(S)wt(S)∑
R∈RPP(sh(S))
xev(R)
Sketch of proof
Given
fα =∑T∈Tα
wt(T )ssh(T )
=∑T∈Tα
wt(T )∑
T ′∈SSYT (sh(T ))
xev(T′)
we want to show
fα =∑S∈Sα
ε(S)wt(S)gsh(S)
=∑S∈Sα
ε(S)wt(S)∑
R∈RPP(sh(S))
xev(R)
The involution
Example
423 41 23 4
21 31 3 4
423 41 23 4
21 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
423 41 23 4
21 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
•21 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
2•1 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21• 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
211 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
211 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21• 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
2•1 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
•21 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
432 41 23 4
21 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
423 41 23 4
21 31 3 4
The involution
Example
423 41 23 4
21 31 3 4
423 41 23 4
21 31 3 4
Sage-Combinat meeting tonight
Sage’s mission:
“To create a viable high-quality and open-source alternative toMapleTM, MathematicaTM, MagmaTM, and MATLABTM”
...“and to foster a friendly community of users and developers”
Tonight, Thorton Hall, Room 326
I 7pm-8pm: Introduction to Sage and Sage-Combinat
I 8pm-10pm: Help on installation & getting startedBring your laptop!
I Design discussions
Thank you for your attention.
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