Using Grassmann calculus in combinatorics: Lindstr … · Grassmann variables calculus Lindstr om-Gessel-Viennot Lemma with cycles A one parameter extension of Schur’s functions

Grassmann variables calculus Lindstrom-Gessel-Viennot Lemma with cycles A one parameter extension of Schur’s functions

Using Grassmann calculus in combinatorics:Lindstrom-Gessel-Viennot lemma and Schur functions

Thomas Krajewski

Centre de Physique Theorique, Marseille

[email protected]

in collaboration with

S. Carrozza and A. Tanasa (LABRI, Bordeaux)

GASCOM 2016

Furiani, June 2-4, 2016


Algebra of Grassmann variables

Definition of a Grassmann algebra

Algebra of Grassmann Λm variables generated by m anticommutingvariables χ1, ..., χm

χiχj = −χjχi , ∀i , j = 1, . . . ,m.

Λm algebra of dimension 2m whose elements are interpreted asfunctions (power series)

f (χ) =m∑

n=0

1

m!

∑1≤i1,... in≤n

ai1...in χi1 . . . χin ,

with antisymmetric coefficients aiσ(1),...,iσ(n) = ε(σ)ai1,...,in .

Multiplication law

(χi1 . . . χin)(χj1 . . . χjp ) =

{0 if{i1, . . . , in} ∩ {j1, . . . , jp} 6= ∅sgn(k)χk1 . . . χkn+p otherwise

with k = (k1, . . . , kn+p) the permutation of (i1, . . . , in, j1, . . . , jp)such that k1 < . . . < kn+p.


Integration in a Grassmann algebra

Definition of Grassman integral

Unique linear form∫dχ =

∫dχm . . . dχ1 on Λm such that∫

dχχi1 . . . χin =

{0 if n < m

sgn(σ) if n = m and ik = σ(k)

Integral of f (χ) =m∑

n=0

1

m!

∑1≤i1,... in≤n

ai1...inχi1 . . . χin ∈ Λm

∫dχ f (χ) = a12...n

Motivated by translational invariance∫dχ f (χ) =

∫dψ g(ψ) with g(ψ) = f (χ) and ψ = χ+ η

.Rules of calculus apply with modifications∫

dχ f (aχ) = a

∫dχ f (χ) (instead of a−1)


Gaussian integral over Grassmann variables

Expression of a determinant as a Grassmann integral

detM =

∫d χNdχN . . . d χ1dχ1 exp

(−

∑1≤i,j≤N

χiMijχj

)d χdχ := d χNdχN . . . d χ1dχ1 integration over 2N Grassmann variables.

Expand the exponential and perform the integration

exp

(−∑

1≤i,j≤N

χiMijχj

)=

∏1≤i,j≤N

exp

(χiMijχj

)=

∏1≤i,j≤N

(1+χiMijχj

)Grassmann version of Gaussian integral∫

dXNdXN . . . dX1dX1 exp

(−

∑1≤i,j≤N

XiMijXj

)=

(2π)N

detM

Extension to minors with lines I = {i1 < · · · < ip} and columnsJ = {j1 < · · · < jp} removed

det(MI cJc ) = (−1)∑

1≤k≤p ik+jk

∫d χdχ χj1 χi1 . . . χjp χip exp

(−∑

1≤i,j≤N

χiMijχj

)


Adjacency and path matrices

G directed graph with N vertices denoted V1, . . . ,Vn with weightswe for edges e from Vi to Vj .

Weighted adjacency matrix Aij =∑

edges e i→j

we

Weighted path matrix Mij =[(1− A)−1

]ij

=∑

paths P i→j

( ∏e∈P

we

)

Example:

V1 V2

V4V3

A =

0 w12 w13 00 0 0 w24

0 0 0 w34

0 0 w23 0

M =

1 w12 (w13 + w12w24w43)C (w34w43) (w13w34 + w12w24)C (w34w43)0 1 w24w43C (w34w43) w24C (w34w43)0 0 C (w34w43) w34C (w34w43)0 0 w43C (w34w43) C (w34w43)

with C (w34w43) =

∞∑k=0

(w34w43)k =1

1− w34w43the contribution of

the cycles 3→ 4→ 3→ . . . (formal power series)


The Lindstrom-Gessel-Viennot Lemma

G directed acyclic graph with weighted path matrix M.

Lindstrom-Gessel-Viennot Lemma

Expression of minors of path matrix as sum over non intersecting paths

detMi1<···<ik | j1<···<jk︸︷︷︸minor

=∑σ∈Sk

non intersecting pathsPl : Vil

→Viσ(l)

ε(σ)∏

1≤l≤k

∏e∈Pk

we

Example:

V1 V2

V4V3

M =

0 w12 w13 + w12w23 + w14w43 + w12w24w43 w14 + w12w24

0 0 w23 + w24w43 w24

0 0 0 00 0 w43 0

detM1,2|3,4 =

w13 + w12w23 + w14w43 + w12w24w43 w14 + w12w24

w23 + w24w43 w24

= w13w24 + w14w43w24 + w12(w24)2w43 + w12w23w24

− w14w23 − w12w24w23 − w14w24w43 − w12(w24)2w43

= w13w24 − w23w14


An extension to graph with cycles

G directed graph with weighted path matrix M (see also K. Talaskahttp://arxiv.org/abs/1202.3128 for a combinatorial proof).

Lindstrom-Gessel-Viennot Lemma for graph with cycles

Expression of minors of path matrix as sum over non intersecting pathsand cycles

detMi1<···<ik | j1<···<jk︸︷︷︸minor

=

∑non intersecting

paths Pl : Vil→ Viσ(l)

and cycles Cs

W (P)W (C)

∑non intersecting cycles Cs

W (C)

with W (P) = (−1)σ∏

1≤l≤k

∏e∈Pl

we and W (C) = (−1)r∏

1≤s≤r

∏e∈Cs

we .

Sketch of the proof:

Write the minor as

∫d χdχ χj1 χi1 . . . χjp χip exp−

{χ(1−M)−1χ

}exp−

{χ(1−M)−1χ

}=

∫d ηdη exp−

{η(1−M)η + ηχ+ χη

}det(1−M)

Expand the exponential and perform the integrations⇒ vertex disjoint degree 1 or 2 subgraphs

http://arxiv.org/abs/1202.3128


An example for a graph with cycles

Example:

V1 V2

V4V3

A =

0 w12 w13 00 0 0 w24

0 0 0 w34

0 0 w23 0

M =

1 w12 (w13 + w12w24w43)C (w34w43) (w13w24 + w12w24)C (w34w43)0 1 w24w43C (w34w43) w24C (w34w43)0 0 C (w34w43) w34C (w34w43)0 0 w43C (w34w43) C (w34w43)

with cycle contribution C (w34w43) =

∞∑k=0

(w34w43)k =1

1− w34w43

detM1,2|3,4 =(w13 + w12w24w43)C (w34w43) (w13w34 + w12w24)C (w34w43)

w24w43C (w34w43) w24C (w34w43)

=w13w24 + w12(w24)2w43 − w13w34w24w43 − w12(w24)2w43(

1− w34w43

)2=

w13w24

1− w34w43


Transfer matrix approach

Discrete time evolution as a sequence of graphs:graph G1 → G2 → · · · → Gn with adjacency matrix

A(i,m),(j,p) :=

wm,i,j if p = m

1 if p = m + 1 and i = j

0 otherwise

with (Am)ij = wm,i,j adjacency matrix of Gm

(acyclic) G1

G2

Gn

Scalar product on Grassmann algebra:

〈f , g〉 =

∫dχdχ exp

(− χχ

)f (χ)g(χ) =

N∑k=0

1

k!

∑1<i1,··· ,ik≤N

ai1...ikbi1...ik

Transfer matrix approach to The Lindstrom-Gessel-Viennot lemma∑non intersecting paths

in G1 → G2 → · · · → GnPl :Vil

∈G1→Vjσ(l)∈Gn

(−1)ε(σ)W (P1) · · ·W (Pk) = 〈j1, .., jk | (1− An)−1 · · · (1− A1)−1︸︷︷︸transfer matrices

|i1, ..., ik〉

Physical interpretation: Path integral for fermionic particles(fermions do not occupy the same state).


Young diagrams and Schur’s functions

Young diagram: λ sequence of r rows of decreasing lengthsλ1 ≥ λ2 ≥ ... ≥ λrSkew Young diagram: λ/µ with µ ≤ λ remove the first µ1 ≤ λ1, ...,µr ≤ λr boxes in λ

Semi Standard (skew) Young Tableau (SSYT) of shape λ/µ: fillλ/µ with integers decreasing along the columns (top to bottom) andnon increasing along the rows (left to right).

Skew Schur function

sλ/µ(x) :=∑

SSYT of shape λ/µ

∏1≤m≤n

xkmm ,

with km = number of times the integer m appears in the SSYT.

2

3 4

2 2

1 3

→ x1(x2)3(x3)2x4


Schur functions and non intersecting lattice paths

Lattice Z2 = · · · → Z→ Z→ · · · oriented from left to right andbottom to top (acyclic)T right translation operator on Z → adjacency matrix Am = xmT

(1− xnT )−1 · · · (1− x1T )−1 =∑k

hk(x)T k

with complete symmetric functions hk(x) =∑

k1+···kn=k

xk11 · · · xknn

Jacobi-Trudi relation from Lindstrom-Gessel-Viennot lemma

sλ(x) =∑

non intersecting lattice paths P1, . . . ,PrPi : (µi−i+l,1)→(λi−i+l,n)

W (P1) · · ·W (Pi ) = det(hλj−µi+i−j(x)

)1≤,i,j≤r

(-4,1) (-2,1) (-1,1) (1,1)

(-3,4) (0,4) (1,4) (2,4)

↔

2

3 4

2 2

1 3


One parameter extension of Schur functions

sλ/µ(x) = 〈λ|U(x)|µ〉 interpreted as a a transition amplitude (seealso P. Zinn-Justin http://arxiv.org/abs/0809.2392) with

U(x) = (1− xnT )−1 · · · (1− x1T )−1

One parameter extension of Schur functions

sλ/µ(a, x) = 〈λ|Ua(x)|µ〉 sλ/µ(1, x) = sλ/µ(x)

Ua(x) =∑k≥0

Sk(a, x)T k with symmetric functions

Sk(a, x) =∑

k1+···kn=k

xk11 . . . xknn∏

1≤m≤n

a(a + 1) . . . (a + km − 1)

km!.

Extended Jacobi-Trudi relation from Lindstrom-Gessel-Viennot lemma

sλ/µ(a, x) = det(Sλj−µi+i−j(a, x)

)1≤i,j≤r .

Example:

s (a, x) = det

(S2(a, x) 1S3(a, x) S2(a, x)

)=

a(a2 − 1)

3

∑1≤m≤n

x3m + a2∑

1≤p<m≤n1≤p≤q≤n

xmxpxq

http://arxiv.org/abs/0809.2392


Convolution identity

Convolution identity for extended Schur functions

sλ/µ(a + b, x) =∑

ν partitionµ≤ν≤λ

sλ/ν(a, x)sν/µ(b, x)

Proof based on interpretation as a transition amplitude

〈λ|Ua+b(x)|µ〉 = 〈λ|Ua(x)Ub(x)|µ〉 =∑

µ≤ν≤λ

〈λ|Ua(x)|ν〉〈ν|Ub(x)|µ〉

Alternative proof based on Cauchy-Binet formula with

Sk(a + b, x) =∑

p+q=k

Sp(a, x)Sq(b, x)

Example: s (a + b, x) = s (a, x) + s (a)s (b)(x)+

s (a, x)s (b, x) + s (a, x)s (b, x) + s (b, x)

Corollary:sλ∗/µ∗(a, x) = (−1)|λ|−|µ|sλ/µ(−a, x)

for the conjugate diagrams (symmetric diagram with respect to themain diagonal)


Vi ringraziu!More on Grassmann variables, Lindstrom-Gessel-Viennot lemma and

Schur functions https://arxiv.org/abs/1604.06276

https://arxiv.org/abs/1604.06276

Documents

Using Grassmann calculus in combinatorics: Lindstr … · Grassmann variables calculus Lindstr om-Gessel-Viennot Lemma with cycles A one parameter extension of Schur’s functions