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Correction to: Math. Ann.
The subdivision map defined in the original version of the paper needs to be modified slightly so that it gives induce an isomorphism for the cyclotomic quotients. The following material replaces the results at the end of section 4D that appears after Example 4D.3.
There is an analogous subdivision map \(S_{\Gamma ,\bar{\Gamma }}^{\prime }\,{:}\,{\mathscr {W}_{{\beta }}^{{\varvec{\rho }}}}(X)\!\longrightarrow \!{\bar{\mathscr {W}}_{{\bar{\beta }}}^{{{\varvec{\rho }}}}(X,\bar{X})}\) given by putting the solid i-string to the left of the ghost s(i)-string, so that (4C.1) becomes:
Everything above works verbatim for this subdivision. The map \(S_{\Gamma ,\bar{\Gamma }}^{\prime }\,{:}\,{\mathscr {W}_{{\beta }}^{{\varvec{\rho }}}}(X)\!\longrightarrow \!{\bar{\mathscr {W}}_{{\bar{\beta }}}^{{{\varvec{\rho }}}}(X,\bar{X})}\) is better adjusted to the cyclotomic quotients:
The isomorphism \(S_{\Gamma ,\bar{\Gamma }}^{\prime }\,{:}\,{\mathscr {W}_{{\beta }}^{{\varvec{\rho }}}}(X)\!\longrightarrow \!{\bar{\mathscr {W}}_{{\bar{\beta }}}^{{{\varvec{\rho }}}}(X,\bar{X})}\) of Theorem 4D.2 descends to an isomorphism of the cyclotomic quotients.
FormalPara ProofThis follows immediately from Theorem 4D.2 since \(S_{\Gamma ,\bar{\Gamma }}^{\prime }\) gives a bijection between unsteady diagrams of \({\mathscr {W}_{{\beta }}^{{\varvec{\rho }}}}(X)\) and the unsteady diagrams of \({\bar{\mathscr {W}}_{{\bar{\beta }}}^{{{\varvec{\rho }}}}(X,\bar{X})}\). \(\square \)
In the infinitesimal-case both Proposition 3F.1 and Theorem 4D.2 apply. Hence, we obtain the following result for the (cyclotomic) KLRW algebra:
We have an isomorphism of graded algebras \(S_{\Gamma ,\bar{\Gamma }}^{\prime }\,{:}\,\mathcal {W}_{{\beta }}^{{{\varvec{\rho }}}}\!\longrightarrow \!\bar{\mathcal {W}_{{\beta }}^{{{\varvec{\rho }}}}}\), where \(\bar{\mathcal {W}_{{\beta }}^{{{\varvec{\rho }}}}}\) is defined as in (4C.13). There are similar isomorphisms for the cyclotomic quotients.
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Mathas, A., Tubbenhauer, D. Correction to: Cellularity and subdivision of KLR and weighted KLRW algebras. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02818-8
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DOI: https://doi.org/10.1007/s00208-024-02818-8