Combinatorics of characters and continuation of Li . V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ o et al. Collaboration at various stages of the work and in the framework of the Project Evolution Equations in Combinatorics and Physics : N. Behr, K. A. Penson, C. Tollu. CIP-CALIN, 18 juin 2019 1/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 1 / 28
Plan 2 Plan practice: property/2 3 Multiplicity Sch¨ utzenberger’s 16 A useful Automaton (Eilenberg, calculus property/3 Sch¨ utzenberger) 9 Examples 18 Properties of the 4 Multiplicity 10 From theory to extended Li The arrow Li (1) automaton (linear practice: construction 20 • representation) & starting from S . 21 Sketch of the behaviour 11 Link with proof for vi. 5 Operations and conc-bialgebras (CAP 23 End of the ladder: definitions on series 17) pushing coefficients to 6 Rational series 12 Link with C C (Sweedler & conc-bialgebras/2 25 Concluding Sch¨ utzenberger) 13 Some dual laws remarks/1 7 Sweedler’s duals 14 A useful property 26 Concluding 8 From theory to 15 A useful remarks/2 2/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 2 / 28
Multiplicity Automaton (Eilenberg, Sch¨ utzenberger) b | α 3 η 1 ν 1 2 3 a | α 1 c | α 5 ν 2 c | α 4 η 2 a | α 9 1 5 b | α 2 c | α 7 4 a | α 8 1 S. Eilenberg, Automata, Languages, and Machines (Vol. A) Acad. Press, New York, 1974 2 M.P. Sch¨ utzenberger, On the definition of a family of automata, Inf. and Contr., 4 (1961) , 245-270. 3/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 3 / 28
Multiplicity automaton (linear representation) & behaviour Linear representation � T � ν 2 0 � � 0 ν = 0 0 η = 0 0 ν 1 , η 1 η 2 0 0 0 0 0 0 0 α 9 α 1 α 2 0 0 0 0 0 0 0 0 0 α 3 µ ( a ) = 0 0 0 0 0 µ ( b ) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 µ ( c ) = 0 0 0 0 α 5 0 0 0 0 α 7 0 0 0 0 α 4 Behaviour �� � � A ( w ) = ν µ ( w ) η = ν ( i ) weight ( p ) η ( j ) i , j � �� � states weight of all paths i � → j � with label w 4/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 4 / 28
Operations and definitions on series � = R X ∗ Addition, Scaling : as for functions because R � � X � Concatenation : f . g ( w ) = � w = uv f ( u ) g ( v ) Polynomials : Series s.t. supp ( f ) = { w } f ( w ) � =0 is finite. The set of polynomials will be denoted R � X � . Pairing : � S | P � = � w ∈ X ∗ S ( w ) P ( w ) ( S series, P polynomial) Summation : � i ∈ I S i summable iff f or all w ∈ X ∗ , i �→ � S i | w � is finitely supported. This corresponds to the product topology (with R discrete). In particular, we have � � � S i := ( � S i | w � ) w i ∈ I w ∈ X ∗ i ∈ I Star : For all series S s.t. � S | 1 X ∗ � = 0, the family ( S n ) n ≥ 0 is summable and we set S ∗ := � n ≥ 0 S n = 1 + S + S 2 + · · · (= (1 − S ) − 1 ). Shifts : � u − 1 S | w � = � S | uw � , � Su − 1 | w � = � S | wu � 5/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 5 / 28
Rational series (Sweedler & Sch¨ utzenberger) Theorem A Let S ∈ k � � X � � TFAE i) The family ( Su − 1 ) u ∈ X ∗ is of finite rank. ii) The family ( u − 1 S ) u ∈ X ∗ is of finite rank. iii) The family ( u − 1 Sv − 1 ) u , v ∈ X ∗ is of finite rank. iv) It exists n ∈ N , λ ∈ k 1 × n , µ : X ∗ → k n × n (a multiplicative morphism) and γ ∈ k n × 1 such that, for all w ∈ X ∗ ( S , w ) = λµ ( w ) γ (1) v) The series S is in the closure of k � X � for (+ , conc , ∗ ) within k � � X � � . Definition A series which fulfill one of the conditions of Theorem A will be called rational . The set of these series will be denoted by k rat � � X � � . 6/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 6 / 28
Sweedler’s duals Remarks 1 (i ↔ iii) needs k to be a field. 2 (iv) needs X to be finite, (iv ↔ v) is known as the theorem of Kleene-Sch¨ utzenberger (M.P. Sch¨ utzenberger, On the definition of a family of automata, Inf. and Contr., 4 (1961) , 245-270.) 3 For the sake of Combinatorial Physics (where the alphabets can be infinite), (iv) has been extended to infinite alphabets and replaced by iv’) The series S is in the rational closure of k X (linear series) within k � � X � � . 4 This theorem is linked to the following: Representative functions on X ∗ (see Eichii Abe, Chari & Pressley), Sweedler’s duals &c. 5 In the vein of (v) expressions like ab ∗ or identities like ( ab ∗ ) ∗ a ∗ = ( a + b ) ∗ (Lazard’s elimination) will be called rational. 7/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 7 / 28
From theory to practice: Sch¨ utzenberger’s calculus From series to automata Starting from a series S , one has a way to construct an automaton (finite-stated iff the series is rational) providing that we know how to compute on shifts and one-letter-shifts will be sufficient due to the formula u − 1 v − 1 S = ( vu ) − 1 S . Calculus on rational expressions In the following, x is a letter, E , F are rational expressions (i.e. expressions built from letters by scalings, concatenations and stars) 1 x − 1 is (left and right) linear 2 x − 1 ( E . F ) = x − 1 ( E ) . F + � E | 1 X ∗ � x − 1 ( F ) 3 x − 1 ( E ∗ ) = x − 1 ( E ) . E ∗ 8/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 8 / 28
Examples With (2 a ) ∗ (3 b ) ∗ ; X = { a , b } a | 2 b | 3 b | 3 1 1 (2 a ) ∗ (3 b ) ∗ (3 b ) ∗ 1 With ( t 2 x 0 x 1 ) ∗ ; X = { x 0 , x 1 } x 0 | t 1 ( t 2 x 0 x 1 ) ∗ tx 1 ( t 2 x 0 x 1 ) ∗ 1 x 1 | t 9/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li .CIP-CALIN, 18 juin 2019 o et al. Collaboration at various stages of the work and in the framework of the 9 / 28
From theory to practice: construction starting from S . States u − 1 S (constructed step by step) Edges We shift every state by letters (length) level by level (knowing that x − 1 ( u − 1 S ) = ( ux ) − 1 S ). Two cases: Returning state : The state is a linear combination of the already created ones i.e. x − 1 ( u − 1 S ) = � v ∈ F α ( ux , v ) v − 1 S (with F finite), then we set the edges x | α v u − 1 S → v − 1 S − The created state is new : Then x | 1 u − 1 S → x − 1 ( u − 1 S ) − Input S with the weight 1 Outputs All states T with weight � T | 1 X ∗ � 10/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li . o et al. Collaboration at various stages of the work and in the framework of the CIP-CALIN, 18 juin 2019 10 / 28
Link with conc-bialgebras (CAP 17) We call here conc-bialgebras, structures such that B = ( k � X � , conc , 1 X ∗ , ∆ , ǫ ) is a bialgebra and ∆( X ) ⊂ ( k . X ⊕ k . 1 X ∗ ) ⊗ 2 . For this, as k � X � is a free algebra, it suffices to define ∆ and check the axioms on letters. Below, some examples Shuffle : X is arbitrary ∆( x ) = x ⊗ 1 + 1 ⊗ x and � ∆( w ) = w [ I ] ⊗ w [ J ] I + J =[1 ···| w | ] Stuffle : Y = { y i } i ≥ 1 , ∆( y k ) = y k ⊗ 1 + 1 ⊗ y k + � i + j = k y i ⊗ y j q -infiltration : X is arbitrary, ∆( x ) = x ⊗ 1 + 1 ⊗ x + q x ⊗ x and � q | I ∩ J | w [ I ] ⊗ w [ J ] ∆( w ) = I ∪ J =[1 ···| w | ] 11/28 V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ Combinatorics of characters and continuation of Li . o et al. Collaboration at various stages of the work and in the framework of the CIP-CALIN, 18 juin 2019 11 / 28
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