Units, central splitting, and filtration for multivariable boxed convolution

Definitions and mathematical statement

Problem 5.2 (A. Nica). Describe the unit group for multivariable boxed convolution, and find a multivariable analogue of the $S$-transform/free Fourier transform. Let $\mathcal S_s={\mathbb{C}}_0\langle\!\langle z_1,\ldots,z_s\rangle\!\rangle$, $s\geq1$, be formal noncommutative series with zero constant coefficient. For a word $w=(i_1,\ldots,i_n)$, write $f_w$ for its coefficient. If $\pi\in\operatorname{NC}(n)$, set $f_{w|\pi}=\prod_{B\in\pi}f_{w|B}$, with each subword read in increasing position order. Using the right Kreweras complement $K(\pi)$, define \[ (f\boxtimes g)_w=\sum_{\pi\in\operatorname{NC}(n)} f_{w|\pi}g_{w|K(\pi)}. \] The complement convention is fixed by the usual maximal noncrossing complement on interleaved unbarred/barred positions. The source's corrupted "?s" denotes this operation. It is associative with unit $\Delta_s=z_1+\cdots+z_s$. For tuples in a unital probability algebra, freeness is between the algebras generated by the two whole tuples; then the source relation is $R_a\boxtimes R_b=R_{(a_1b_1,\ldots,a_sb_s)}$. No internal independence within each tuple is assumed.

Reported units and central splitting. A formal series is invertible exactly when \[ f_{(i)}\ne0\qquad(1\leq i\leq s). \] Its inverse is determined recursively by word length. Explicitly, if $a_i=f_{(i)}$, $a_w=\prod_ja_{i_j}$, the right-inverse coefficients start at $g_{(i)}=a_i^{-1}$, and for $|w|\geq2$ satisfy \[ g_w=-a_w^{-1}\left[f_wa_w^{-1} +\sum_{\pi\ne0_n,1_n}f_{w|\pi}g_{w|K(\pi)}\right]. \] All coefficients in the sum have smaller word length; a left recursion gives the same group inverse.

For $a\in({\mathbb{C}}^*)^s$, let $T_a=\sum_i a_iz_i$. The aid reports the coefficientwise central action \[ (T_a\boxtimes f)_w=(f\boxtimes T_a)_w=a_wf_w. \] If $U_s$ is the subgroup with all linear coefficients one, every unit has the unique factorization $f=T_a\boxtimes u$, $u\in U_s$, hence the canonical direct product \[ \mathcal S_s^\times\cong({\mathbb{C}}^*)^s\times U_s. \] This is a formal scalar-coefficient statement, not an analytic or operator-valued transform theorem.

For $p\geq2$, define $F^pU_s$ by vanishing of all nonlinear coefficients of degrees less than $p$. The reported commutator estimate is \[ [F^pU_s,F^qU_s]\subseteq F^{p+q-1}U_s\qquad(p,q\geq2). \] Finite degree truncations are nilpotent unipotent algebraic groups, and $U_s$ is their degree-complete inverse limit. The aid reports a first nonzero graded commutator in degree three, proving noncommutativity for $s\geq2$; its complete bracket formula is not included in the supplied summary. Consequently no injective group homomorphism of this full unit group into an abelian multiplicative target can provide a faithful multivariable transform. Noncommutative target constructions are not excluded or completed by this obstruction.

One-variable source remarks and target correction. For $s=1$, boxed convolution is commutative, and the source cites Nica--Speicher's free Fourier isomorphism $F$, satisfying $F(R_a)=S_a$ when $\varphi(a)\ne0$. With the above cumulant-series convention, the usual expression is $F(f)(z)=f^{\langle-1\rangle}(z)/z$, using compositional inverse. Its multiplicative target must be ${\mathbb{C}}[[z]]^\times$, the series with nonzero constant coefficient. The literal source target ${\mathbb{C}}_0[[z]]$ under ordinary multiplication has no unit and cannot have the claimed unit group. This is an explicit notation defect, not a silent redefinition. The requested multivariable $S$-transform remains broader than the supplied group-structure result.

Short English statement

The aid describes boxed-convolution units by nonzero linear coefficients and splits their group into a central torus and a normalized inverse limit of nilpotent groups. Multivariable noncommutativity blocks faithful transforms into abelian multiplicative targets. The one-variable remark also needs a target correction: ordinary multiplicative units have nonzero constant term.

Sources

• Local UnsolvedMath registry, permanent ID $20002720$, catalogue entry, aim-probability-notes.json, notes[161], workshop Braid groups, clusters and free probability, Problem 5.2, A. Nica: https://aimath.org/WWN/braidgroups/braidgroups.pdf.
• Original references [39], Alexandru Nica and Roland Speicher, multivariable boxed convolution/noncrossing partitions; [38], their free Fourier transform; the remarks also refer to Voiculescu's $S$-transform. Full bibliography entries are not supplied.
• Supplied research summary on inverse recursions, central torus, degree filtration and first graded bracket. Attempt 1: partial_result, partially_solved, rigor "proved", low novelty confidence. The machine-generated aid is not peer reviewed and requires independent verification before citation. The supplied verification identifies Problem 5.2; this draft has not visited the PDF. The extracted 11 is a page marker.