Cumulants, Hausdor ff Series, and Quasisymmetric Functions T. - PowerPoint PPT Presentation

Cumulants, Hausdor ff Series, and Quasisymmetric Functions T. Hasebe, F. Lehner, J.-C. Novelli, J.-Y. Thibon https://arxiv.org/abs/1711.00219 https://arxiv.org/abs/2006.02089

Classical cumulants . Let m n “ m n p X q “ E X n be the moments of a random variable X . The cumulants are characterized by the following properties (K1) Additivity : If X and Y are independent random variables, then κ n p X ` Y q “ κ n p X q ` κ n p Y q . (K2) Homogeneity : For any scalar λ the n -th cumulant is n -homogeneous: κ n p λ X q “ λ n κ n p X q . (K3) Universality : There exist universal polynomials P n in n ´ 1 variables without constant term such that m n p X q “ κ n p X q ` P n p κ 1 p X q , κ 2 p X q , ..., κ n ´ 1 p X qq .

Generating function . The exponential generating functions satisfy the identity De fi nition. 8 8 m n κ n n ! t n “ exp ÿ ÿ n ! t n n “ 0 n “ 1 Thiele (1889): “halvinvarianter”, Hausdor ff (1901): “logarithmische Momente”

Symmetric functions . compare with symmetric functions ˜ 8 8 ¸ p n p X q h n p X q t n “ exp ÿ ÿ t n H t p X q “ n n “ 0 n “ 1

Character . χ X p h n q “ m n p X q n !

Coproducts . ∆ f p X, Y q “ f p X Y Y q “ : f p X ` Y q δ f p X, Y q “ f p X ˆ Y q “ : f p XY q p Sym , ¨ , ∆ q is a Hopf algebra.

Formalization of independence . Let p A , ϕ q be a ncps. X and Y are independent if ϕ p XY q “ ϕ p X q ϕ p Y q or formally d p X, Y q » p X b 1 , 1 b Y q in p A b A , ϕ b ϕ q

Algebraic setup . For a given ncps p A , ϕ q let U “ A b8 ϕ b8 ϕ “ ˜ ˜ and embed X ÞÑ X p i q “ I b I b ¨ ¨ ¨ I b X b I b ¨ ¨ ¨ d Similarly, X and Y are free , Boolean independent etc., if p X, Y q » p X p 1 q , Y p 2 q q where U “ A ˚ 8 free product, etc.

Lattice reformulation of independence . De fi nition. Subalgebras A j are independent if ϕ π p X 1 , X 2 , . . . , X n q “ ϕ π ^ η p X 1 , X 2 , . . . , X n q whenever η is a partition of X i such that the X i from each block come from one of the subalgebras and subalgebras for di ff erent blocks are di ff erent.

Cumulants . Rota’s dot operation N.X “ X p 1 q ` X p 2 q ` ¨ ¨ ¨ ` X p N q ϕ pp N.X 1 qp N.X 2 q ¨ ¨ ¨ p N.X n qq “ N ¨ K n p X 1 , X 2 , . . . , X n q ` ω p N 2 q ˜

Partitioned cumulants and M¨ obius inversion . ϕ π p N.X 1 , N.X 2 , . . . , N.X n q “ N | π | K π p X 1 , X 2 , . . . , X n q ` ω p N | π | ` 1 q Theorem. ÿ K π p X 1 , X 2 , . . . , X n q “ ϕ σ p X 1 , X 2 , . . . , X n q µ p σ , π q σ ď π

Mixed cumulants . Theorem. Independence ð ñ mixed cumulants vanish.

Weisner’s Lemma (1935) P a lattice, a, b, c P P , then # µ p b, c q a ě c ÿ µ p x, c q “ 0 otherwise x P P x ^ a “ b

NC symmetric functions . X “ t X 1 , X 2 , . . . u noncommutative alphabet. WSym is the algebra generated by the monomial symmetric func- tions ÿ m π “ X i 1 X i 2 . . . X i n ker i “ π

NC power sums . ÿ φ π “ X i 1 X i 2 . . . X i n ker i ě π ÿ “ m σ σ ě π ÿ m π “ µ p π , σ q φ σ σ ě π

Calculation rules . ÿ m π m ρ “ m σ σ ^p ˆ 1 m | ˆ 1 n q“ π | ρ φ π φ σ “ φ π | σ

Coproduct . As before the coproduct ∆ F p X, Y q “ ∆ F p X ` Y q is cocommutative.

Dual basis . The dual WSym ˚ is commutative. De fi ne dual bases N π and Φ π by x N π , m σ y “ δ π , σ x Φ π , φ σ y “ δ π , σ Then N π “ ÿ Φ σ σ ď π Φ π “ ÿ N σ µ p σ , π q σ ď π

“Character” . Given a sequence p X i q Ď A , we de fi ne a linear map ϕ : WSym ˚ : Ñ C ˆ ϕ p N π q “ ϕ π p X 1 , X 2 , . . . , X n q ˆ Then cumulants are encoded by ϕ p Φ π q “ K π p X 1 , X 2 , . . . , X n q ˆ

Internal product . The internal product on WSym ˚ is inherited from WQSym ˚ (later) and takes the form N π ˚ N σ “ N π ^ σ and thus is an incarnation of the M¨ obius algebra of the partition lattice.

M¨ obius idempotents . e π : “ ř σ ď π σ µ p σ , π q are orthogonal idempotents in the M¨ obius alge- bra Z r Π n s and thus Φ π “ ÿ N σ µ p σ , π q σ ď π are orthogonal idempotents in WSym ˚ with respect to the internal product.

Independence and mixed cumulants revisited . Whenever η P Π n is a partition of X i into mutually independent, then ϕ p N π ˚ N η q ϕ p N π ^ η q “ ˆ ϕ p N π q “ ˆ ˆ and thus ϕ p Φ π ˚ N η q “ 0 K π p X 1 , X 2 , . . . , X n q “ ˆ ϕ p Φ π q “ ˆ because for π ę η we have Φ π ˚ N η “ 0

Spreadability and ordered set partitions . A spreadability system is an algebra p U , ˜ ϕ q and a family of em- beddings A Ñ U X ÞÑ X p i q such that ϕ p X p i 1 q X p i 2 q ϕ p X p h p i 1 qq X p h p i 2 qq ¨ ¨ ¨ X p i n q ¨ ¨ ¨ X p h p i n qq ˜ q “ ˜ q n n 1 2 1 2 for every strictly increasing map h : N Ñ N .

Packed words . A packed word is a word w “ w 1 w 2 . . . w n , with w i P N such that no letter is left out, i.e., if k occurs, then all l ă k occur as well. Packed words encode ordered set partitions. Any word can be arranged into a packed word pack p w q » ker w i.e., if b 1 ă b 2 ă ¨ ¨ ¨ ă b k are the letters occuring in w , then pack p w q is obtained by replacing each b j by j .

Independence . X and Y are independent if d » p X p 1 q , Y p 2 q q p X, Y q d » p X p 2 q , Y p 1 q q ). (but not necessarily p X, Y q Partitioned moments φ π or “packed moments” φ u are analogously.

Quasisymmetric functions . Let X “ X 1 , X 2 , . . . be an (in fi nite) alphabet. A quasisymmetric function is a formal power series ÿ f p x 1 , x 2 , . . . q “ x α 1 i 1 x α 2 i 2 ¨ ¨ ¨ x α k i k such that the coe ffi cients are invariant under spreadings only: r x α 1 i 1 x α 2 i 2 ¨ ¨ ¨ x α n i n s f “ r x α 1 1 x α 2 2 ¨ ¨ ¨ x α n n s f for any sequence i 1 ă i 2 ă ¨ ¨ ¨ ă i n .

NC Quasisymmetric functions . Let X “ X 1 , X 2 , . . . be an (in fi nite) noncommuting alphabet. The algebra WQSym of noncommutative (word) quasisym- metric functions is spanned by the “monomials” ÿ M u “ X w pack p w q“ u Again we de fi ne ∆ p f q “ f p X ‘ Y q δ p f q “ f p X ˆ Y q where X ‘ Y is the ordered sum of alphabets and X ˆ Y carries the lexicographic order.

Duality . WQSym is noncommutative and non-cocommutative. Let N u be the dual basis of M u x N u , M v y “ δ u,v We de fi ne as before ϕ p N u q “ ϕ π p X 1 , X 2 , . . . , X n q ˆ where π “ ker u (ordered kernel).

Solomon-Tits algebra . Let η be an ordered partition of X i into mutually independent sub- sets, then φ π p X 1 , X 2 , . . . , X n q “ φ π N η p X 1 , X 2 , . . . , X n q where π N η “ p π 1 X η 1 , π 1 X η 2 , . . . q i.e., intersection π i X η j in lexicographic order. In terms of packed words this is the internal product on WQSym ˚ (induced by δ ), which is isomorphic to the Solomon-Tits algebra .

Cumulants . Cumulants are de fi ned as before ϕ pp N.X 1 qp N.X 2 q ¨ ¨ ¨ p N.X n qq “ N ¨ K n p X 1 , X 2 , . . . , X n q ` ω p N 2 q ˜

Factorial M¨ obius inversion . Theorem. ÿ ϕ π p X 1 , X 2 , . . . , X n q ˜ ϕ π p X 1 , X 2 , . . . , X n q “ ζ p σ , π q σ ď π ÿ K π p X 1 , X 2 , . . . , X n q “ ϕ π p X 1 , X 2 , . . . , X n q ˜ µ p σ , π q σ ď π where 1 ˜ ζ p σ , ˆ 1 q “ | σ | ! 1 q “ p´ 1 q | σ | ´ 1 µ p σ , ˆ ˜ | σ | σ , ˆ “ µ p ¯ 1 q | σ | !

Eulerian idempotents . If we set as before ϕ p N u q “ ϕ π p X 1 , X 2 , . . . , X n q ˆ where π “ ker u , then ϕ p N u ˚ E r r s K π p X 1 , X 2 , . . . , X n q “ ˆ n q where r “ | π | and E r r s is the so-called Euler idempotent . n

Mixed cumulants . Theorem. Whenever X i can be partitioned into mutually indepen- dent subsets, say into η P Π n , then ÿ K n p X 1 , X 2 , . . . , X n q “ K τ p X 1 , X 2 , . . . , X n q g p τ , η q τ where g p τ , η q are the Goldberg coe ffi cients appearing in the Campbell- Baker-Hausdor ff series.