$$
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$$
E0052 definition
Shuffle coproduct
The symmetric algebra \(\SS(V)\) E0051 carries a cocommutative coproduct that makes it a bialgebra.
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The shuffle coproduct \(\Delta: \SS(V) \to \SS(V) \otimes \SS(V)\) is the unique algebra homomorphism with \(\Delta(v) = v \otimes 1 + 1 \otimes v\) for \(v \in V\).
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On decomposables:
\[ \Delta(v_1 \cdots v_k) = \sum_{I \sqcup J = [k]} v_I \otimes v_J. \] -
The reduced coproduct \(\bar{\Delta}: \SS_+(V) \to \SS_+(V) \otimes \SS_+(V)\) restricts to nonempty \(I, J\).
-
Higher coproducts: \(\Delta^{(r-1)}(v_1 \cdots v_k) = \sum_{\pi \in \Part^{\mathrm{ord}}(k, r)} \bigotimes_{B \in \pi} v_B\).
(2) Proposition (Conilpotence). \(\bar{\Delta}\) strictly lowers degree: \(\bar{\Delta}(\SS^{\leq k}) \subseteq \SS^{\leq k-1} \otimes \SS^{\leq k-1}\), hence \(\bar{\Delta}^k = 0\) on \(\SS^{\leq k}\).
Proof. Each summand \(v_I \otimes v_J\) of \(\bar{\Delta}(v_1 \cdots v_k)\) has \(|I|, |J| \geq 1\) and \(|I| + |J| = k\), so both factors have degree \(\leq k - 1\). Iterating \(m\) times, each tensor factor has degree \(\leq k - m\); at \(m = k\) every factor has degree \(\leq 0\), but \(\bar{\Delta}\) projects away degree \(0\), giving \(0\).
(3) Proposition (Bialgebra compatibility). \(\Delta(a \cdot b) = \Delta(a) \cdot \Delta(b)\). Multiplication and coproduct are adjoint under the permanent pairing E0053: \((\alpha \cdot \beta, a) = (\alpha \otimes \beta, \Delta a)\).
Proof. Multiplicativity holds by definition: \(\Delta\) is an algebra homomorphism. For the adjunction, expand \((\alpha_1 \cdots \alpha_r, a \cdot b)\) by the permanent: each permutation \(\sigma \in S_r\) splits into a part acting on the \(a\)-factors and a part on the \(b\)-factors, corresponding to a shuffle decomposition \(I \sqcup J = [r]\). Collecting gives \(\sum_{I \sqcup J} (\alpha_I, a)(\alpha_J, b) = (\alpha \otimes \alpha, \Delta(a \cdot b)) = (\alpha \cdot \beta, a) \cdot (\text{etc.})\). The identity \((\alpha \cdot \beta, a) = (\alpha \otimes \beta, \Delta a)\) follows by the same shuffle argument applied to \(\Delta a\).
Used by: