Grassmann variables calculus Lindstr¨ om-Gessel-Viennot Lemma with cycles A one parameter extension of Schur’s functions Using Grassmann calculus in combinatorics: Lindstr¨ om-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
Grassmann variables calculus Lindstr¨ om-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 Λ m variables generated by m anticommuting variables χ 1 , ..., χ m χ i χ j = − χ j χ i , ∀ i , j = 1 , . . . , m . Λ m algebra of dimension 2 m whose elements are interpreted as functions (power series) m 1 � � f ( χ ) = a i 1 ... i n χ i 1 . . . χ i n , m ! n =0 1 ≤ i 1 ,... i n ≤ n with antisymmetric coefficients a i σ (1) ,..., i σ ( n ) = ǫ ( σ ) a i 1 ,..., i n . Multiplication law � 0 if { i 1 , . . . , i n } ∩ { j 1 , . . . , j p } � = ∅ ( χ i 1 . . . χ i n )( χ j 1 . . . χ j p ) = sgn ( k ) χ k 1 . . . χ k n + p otherwise with k = ( k 1 , . . . , k n + p ) the permutation of ( i 1 , . . . , i n , j 1 , . . . , j p ) such that k 1 < . . . < k n + p .
Grassmann variables calculus Lindstr¨ om-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 d χ = d χ m . . . d χ 1 on Λ m such that � � 0 if n < m d χ χ i 1 . . . χ i n = sgn ( σ ) if n = m and i k = σ ( k ) m 1 � � Integral of f ( χ ) = a i 1 ... i n χ i 1 . . . χ i n ∈ Λ m m ! n =0 1 ≤ i 1 ,... i n ≤ n � d χ f ( χ ) = a 12 ... n Motivated by translational invariance � � d χ f ( χ ) = d ψ g ( ψ ) with g ( ψ ) = f ( χ ) and ψ = χ + η . Rules of calculus apply with modifications � � (instead of a − 1 ) d χ f ( a χ ) = a d χ f ( χ )
Grassmann variables calculus Lindstr¨ om-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 ¯ χ N d χ N . . . d ¯ χ 1 d χ 1 exp − χ i M ij χ j ¯ 1 ≤ i , j ≤ N d ¯ χ d χ := d ¯ χ N d χ N . . . d ¯ χ 1 d χ 1 integration over 2 N Grassmann variables. Expand the exponential and perform the integration � � � � � � � � � − exp χ i M ij χ j ¯ = exp χ i M ij χ j ¯ = 1+¯ χ i M ij χ j 1 ≤ i , j ≤ N 1 ≤ i , j ≤ N 1 ≤ i , j ≤ N Grassmann version of Gaussian integral � � � = (2 π ) N � d ¯ X N dX N . . . d ¯ ¯ − X 1 dX 1 exp X i M ij X j det M 1 ≤ i , j ≤ N Extension to minors with lines I = { i 1 < · · · < i p } and columns J = { j 1 < · · · < j p } removed � � � � � 1 ≤ k ≤ p i k + j k det( M I c J c ) = ( − 1) d ¯ χ d χ χ j 1 ¯ χ i 1 . . . χ j p ¯ χ i p exp − χ i M ij χ j ¯ 1 ≤ i , j ≤ N
Grassmann variables calculus Lindstr¨ om-Gessel-Viennot Lemma with cycles A one parameter extension of Schur’s functions Adjacency and path matrices G directed graph with N vertices denoted V 1 , . . . , V n with weights w e for edges e from V i to V j . � Weighted adjacency matrix A ij = w e edges e i → j � � � � � (1 − A ) − 1 � Weighted path matrix M ij = ij = w e e ∈P paths P i → j 0 w 12 w 13 0 V 3 V 4 0 0 0 w 24 Example: A = 0 0 0 w 34 0 0 w 23 0 V 1 V 2 1 ( w 13 + w 12 w 24 w 43 ) C ( w 34 w 43 ) ( w 13 w 34 + w 12 w 24 ) C ( w 34 w 43 ) w 12 0 1 w 24 w 43 C ( w 34 w 43 ) w 24 C ( w 34 w 43 ) M = 0 0 C ( w 34 w 43 ) w 34 C ( w 34 w 43 ) 0 0 w 43 C ( w 34 w 43 ) C ( w 34 w 43 ) ∞ 1 � ( w 34 w 43 ) k = with C ( w 34 w 43 ) = the contribution of 1 − w 34 w 43 k =0 the cycles 3 → 4 → 3 → . . . (formal power series)
Grassmann variables calculus Lindstr¨ om-Gessel-Viennot Lemma with cycles A one parameter extension of Schur’s functions The Lindstr¨ om-Gessel-Viennot Lemma G directed acyclic graph with weighted path matrix M . Lindstr¨ om-Gessel-Viennot Lemma Expression of minors of path matrix as sum over non intersecting paths � � � det M i 1 < ··· < i k | j 1 < ··· < j k = ǫ ( σ ) w e � �� � σ ∈ S k 1 ≤ l ≤ k e ∈P k minor non intersecting paths P l : V il → V i σ ( l ) Example: 0 w 12 w 13 + w 12 w 23 + w 14 w 43 + w 12 w 24 w 43 w 14 + w 12 w 24 V 3 V 4 0 0 w 23 + w 24 w 43 w 24 M = 0 0 0 0 0 0 w 43 0 V 1 V 2 det M 1 , 2 | 3 , 4 = w 13 + w 12 w 23 + w 14 w 43 + w 12 w 24 w 43 w 14 + w 12 w 24 w 23 + w 24 w 43 w 24 = w 13 w 24 + w 14 w 43 w 24 + w 12 ( w 24 ) 2 w 43 + w 12 w 23 w 24 − w 14 w 23 − w 12 w 24 w 23 − w 14 w 24 w 43 − w 12 ( w 24 ) 2 w 43 = w 13 w 24 − w 23 w 14
Grassmann variables calculus Lindstr¨ om-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¨ om-Gessel-Viennot Lemma for graph with cycles Expression of minors of path matrix as sum over non intersecting paths and cycles � W ( P ) W ( C ) non intersecting paths P l : V il → V i σ ( l ) and cycles C s det M i 1 < ··· < i k | j 1 < ··· < j k = � W ( C ) � �� � non intersecting cycles C s minor � � � � with W ( P ) = ( − 1) σ w e and W ( C ) = ( − 1) r w e . 1 ≤ l ≤ k e ∈P l 1 ≤ s ≤ r e ∈C s Sketch of the proof: � � � χ (1 − M ) − 1 χ Write the minor as d ¯ χ d χ χ j 1 ¯ χ i 1 . . . χ j p ¯ χ i p exp − ¯ � � � d ¯ η d η exp − ¯ η (1 − M ) η + ¯ ηχ + ¯ χη � � χ (1 − M ) − 1 χ exp − ¯ = det(1 − M ) Expand the exponential and perform the integrations ⇒ vertex disjoint degree 1 or 2 subgraphs
Grassmann variables calculus Lindstr¨ om-Gessel-Viennot Lemma with cycles A one parameter extension of Schur’s functions An example for a graph with cycles 0 w 12 w 13 0 V 3 V 4 0 0 0 w 24 Example: A = 0 0 0 w 34 0 0 w 23 0 V 1 V 2 1 w 12 ( w 13 + w 12 w 24 w 43 ) C ( w 34 w 43 ) ( w 13 w 24 + w 12 w 24 ) C ( w 34 w 43 ) 0 1 w 24 w 43 C ( w 34 w 43 ) w 24 C ( w 34 w 43 ) M = 0 0 C ( w 34 w 43 ) w 34 C ( w 34 w 43 ) 0 0 w 43 C ( w 34 w 43 ) C ( w 34 w 43 ) ∞ 1 � ( w 34 w 43 ) k = with cycle contribution C ( w 34 w 43 ) = 1 − w 34 w 43 k =0 det M 1 , 2 | 3 , 4 = ( w 13 + w 12 w 24 w 43 ) C ( w 34 w 43 ) ( w 13 w 34 + w 12 w 24 ) C ( w 34 w 43 ) w 24 w 43 C ( w 34 w 43 ) w 24 C ( w 34 w 43 ) = w 13 w 24 + w 12 ( w 24 ) 2 w 43 − w 13 w 34 w 24 w 43 − w 12 ( w 24 ) 2 w 43 � � 2 1 − w 34 w 43 w 13 w 24 = 1 − w 34 w 43
Grassmann variables calculus Lindstr¨ om-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 G 1 → G 2 → · · · → G n with adjacency matrix G n w m , i , j if p = m A ( i , m ) , ( j , p ) := 1 if p = m + 1 and i = j 0 otherwise G 2 with ( A m ) ij = w m , i , j adjacency matrix of G m G 1 (acyclic) Scalar product on Grassmann algebra: � N 1 � � � � � f , g � = d χ d χ exp − χχ f ( χ ) g ( χ ) = a i 1 ... i k b i 1 ... i k k ! k =0 1 < i 1 , ··· , i k ≤ N Transfer matrix approach to The Lindstr¨ om-Gessel-Viennot lemma � ( − 1) ǫ ( σ ) W ( P 1 ) · · · W ( P k ) = � j 1 , .., j k | (1 − A n ) − 1 · · · (1 − A 1 ) − 1 | i 1 , ..., i k � � �� � non intersecting paths transfer matrices in G 1 → G 2 → · · · → G n P l : V il ∈ G 1 → V j σ ( l ) ∈ G n Physical interpretation: Path integral for fermionic particles (fermions do not occupy the same state).
Grassmann variables calculus Lindstr¨ om-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 λ 1 ≥ λ 2 ≥ ... ≥ λ r Skew Young diagram: λ/µ with µ ≤ λ remove the first µ 1 ≤ λ 1 , ... , µ r ≤ λ r 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 � � x k m s λ/µ ( x ) := m , 1 ≤ m ≤ n SSYT of shape λ/µ with k m = number of times the integer m appears in the SSYT. 1 3 2 2 x 1 ( x 2 ) 3 ( x 3 ) 2 x 4 → 3 4 2
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