[PPT] - Using Grassmann calculus in combinatorics: Lindstr PowerPoint Presentation

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions

Using Grassmann calculus in combinatorics: Lindstr¨

m-Gessel-Viennot lemma and Schur functions

Thomas Krajewski Centre de Physique Th´ eorique, Marseille krajew@cpt.univ-mrs.fr in collaboration with

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

GASCOM 2016 Furiani, June 2-4, 2016

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions Algebra of Grassmann variables

Definition of a Grassmann algebra Algebra of Grassmann Λm variables generated by m anticommuting variables χ1, ..., χm χiχj = −χjχi, ∀i, j = 1, . . . , m. Λm algebra of dimension 2m whose elements are interpreted as functions (power series) f (χ) =

m

n=0

1 m!

1≤i1,... in≤n

ai1...in χi1 . . . χin, with antisymmetric coefficients aiσ(1),...,iσ(n) = ǫ(σ)ai1,...,in. Multiplication law (χi1 . . . χin)(χj1 . . . χjp) =

if{i1, . . . , in} ∩ {j1, . . . , jp} = ∅

sgn(k) χk1 . . . χkn+p

therwise

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

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions Integration in a Grassmann algebra

Definition of Grassman integral Unique linear form

dχ =
dχm . . . dχ1 on Λm such that
dχ χi1 . . . χin =
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)

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions Gaussian integral over Grassmann variables

Expression of a determinant as a Grassmann integral det M =

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
d ¯

XNdXN . . . d ¯ X1dX1 exp

−
1≤i,j≤N

¯ XiMijXj

= (2π)N

det M Extension to minors with lines I = {i1 < · · · < ip} and columns J = {j1 < · · · < jp} removed det(MI cJc) = (−1)

1≤k≤p ik+jk
d ¯

χdχ χj1 ¯ χi1 . . . χjp ¯ χip exp

−
1≤i,j≤N

¯ χiMijχj

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions Adjacency and path matrices

G directed graph with N vertices denoted V1, . . . , Vn with weights we 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 V4 V3

A =     w12 w13 w24 w34 w23     M =     1 w12 (w13 + w12w24w43)C(w34w43) (w13w34 + w12w24)C(w34w43) 1 w24w43C(w34w43) w24C(w34w43) C(w34w43) w34C(w34w43) w43C(w34w43) C(w34w43)     with C(w34w43) =

∞

k=0

(w34w43)k = 1 1 − w34w43 the contribution of the cycles 3 → 4 → 3 → . . . (formal power series)

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions The Lindstr¨

m-Gessel-Viennot Lemma

G directed acyclic graph with weighted path matrix M. Lindstr¨

m-Gessel-Viennot Lemma

Expression of minors of path matrix as sum over non intersecting paths det Mi1<···<ik| j1<···<jk

minor

=

σ∈Sk

non intersecting paths Pl: Vil →Viσ(l)

ǫ(σ)

1≤l≤k
e∈Pk

we Example:

V1 V2 V4 V3

M =     w12 w13 + w12w23 + w14w43 + w12w24w43 w14 + w12w24 w23 + w24w43 w24 w43     det M1,2|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

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions An extension to graph with cycles

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

m-Gessel-Viennot Lemma for graph with cycles

Expression of minors of path matrix as sum over non intersecting paths and cycles det Mi1<···<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

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions An example for a graph with cycles

Example:

V1 V2 V4 V3

A =     w12 w13 w24 w34 w23     M =     1 w12 (w13 + w12w24w43)C(w34w43) (w13w24 + w12w24)C(w34w43) 1 w24w43C(w34w43) w24C(w34w43) C(w34w43) w34C(w34w43) w43C(w34w43) C(w34w43)     with cycle contribution C(w34w43) =

∞

k=0

(w34w43)k = 1 1 − w34w43 det M1,2|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

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions 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

therwise

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 Lindstr¨

m-Gessel-Viennot lemma
non intersecting paths

in G1 → G2 → · · · → Gn Pl: Vil ∈G1→Vjσ(l)∈Gn

(−1)ǫ(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).

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions Young diagrams and Schur’s functions

Young diagram: λ sequence of r rows of decreasing lengths λ1 ≥ λ2 ≥ ... ≥ λr Skew 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: fill λ/µ with integers decreasing along the columns (top to bottom) and non increasing along the rows (left to right). Skew Schur function sλ/µ(x) :=

SSYT of shape λ/µ
1≤m≤n

xkm

m ,

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

2 3 4 2 2 1 3

→ x1(x2)3(x3)2x4

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions Schur functions and non intersecting lattice paths

Lattice Z2 = · · · → Z → Z → · · · oriented from left to right and bottom 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

xk1

1 · · · xkn n

Jacobi-Trudi relation from Lindstr¨

m-Gessel-Viennot lemma

sλ(x) =

non intersecting lattice paths P1, . . . , Pr

Pi : (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

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions One parameter extension of Schur functions

sλ/µ(x) = λ|U(x)|µ interpreted as a a transition amplitude (see also 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

xk1

1 . . . xkn n

1≤m≤n

a(a + 1) . . . (a + km − 1) km! . Extended Jacobi-Trudi relation from Lindstr¨

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

= a(a2 − 1)

3

1≤m≤n

x3

m + a2

1≤p<m≤n 1≤p≤q≤n

xmxpxq

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions Convolution identity

Convolution identity for extended Schur functions sλ/µ(a + b, x) =

ν partition

µ≤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)|νν|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 the main diagonal)

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Grassmann variables calculus Lindstr¨

m-Gessel-Viennot Lemma with cycles

A one parameter extension of Schur’s functions

Using Grassmann calculus in combinatorics: Lindstr - - PowerPoint PPT Presentation

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