Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

Adrian Celestino, Kurusch Ebrahimi-Fard, Alexandru Nica, Daniel Perales, Leon Witzman

Abstract: We consider the group (G,⁎)<math><mo stretchy="false" is="true">(</mo><mi mathvariant="script" is="true">G</mi><mo is="true">,</mo><mo is="true">⁎</mo><mo stretchy="false" is="true">)</mo></math> of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where “⁎” denotes the convolution operation. We introduce a larger group (G˜,⁎)<math><mo stretchy="false" is="true">(</mo><mover accent="true" is="true"><mrow is="true"><mi mathvariant="script" is="true">G</mi></mrow><mrow is="true"><mo is="true">˜</mo></mrow></mover><mo is="true">,</mo><mo is="true">⁎</mo><mo stretchy="false" is="true">)</mo></math> of unitized functions from the same incidence algebra, which satisfy a weaker semi-multiplicativity condition. The natural action of G˜<math><mover accent="true" is="true"><mrow is="true"><mi mathvariant="script" is="true">G</mi></mrow><mrow is="true"><mo is="true">˜</mo></mrow></mover></math> on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of G˜<math><mover accent="true" is="true"><mrow is="true"><mi mathvariant="script" is="true">G</mi></mrow><mrow is="true"><mo is="true">˜</mo></mrow></mover></math> in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of t-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of Bożejko and Wysoczanski.It is known that the group G<math><mi mathvariant="script" is="true">G</mi></math> can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that G˜<math><mover accent="true" is="true"><mrow is="true"><mi mathvariant="script" is="true">G</mi></mrow><mrow is="true"><mo is="true">˜</mo></mrow></mover></math> can also be identified as group of characters of a Hopf algebra T<math><mi mathvariant="script" is="true">T</mi></math>, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion G⊆G˜<math><mi mathvariant="script" is="true">G</mi><mo is="true">⊆</mo><mover accent="true" is="true"><mrow is="true"><mi mathvariant="script" is="true">G</mi></mrow><mrow is="true"><mo is="true">˜</mo></mrow></mover></math> turns out to be the dual of a natural bialgebra homomorphism from T<math><mi mathvariant="script" is="true">T</mi></math> onto Sym.