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9. Miscellaneous


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      Problem 9.1.

      What is the action of the full twist on the trace of the Hecke category? Does it relate to $\nabla$? How does it act on Schur objects?


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          Problem 9.2.

          Let $B_w\in$ SBim$_n$ be the indecomposable Soergel bimodule associated to $w\in S_n$. Compute Ext$_{R-R \text{ bimod}}(B_v,B_w)$ and develop its diagrammatics.


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              Problem 9.3.

              Find functors from derived horizontal trace computing $\mathcal{gl}(m|n)$-homology.


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                  Problem 9.4.

                  Find a basis of the coinvariant ring of $$\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n,\theta_1,\ldots,\theta_n,\phi_1,\ldots,\phi_n]$$ and a combinatorial model for the Frobenius characteristic in the $$\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n,\theta_1,\ldots,\theta_n]$$ case, does the basis proposed by Haglund-Sergel work?


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                      Problem 9.5.

                      Take $P,Q\in SYT(\lambda)$, how do $tr(P)$ and $tr(G)$ relate?


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                          Problem 9.6.

                          1. Relate $tr(T(m,n))$ to rings of quasi-invariants and to A. Wilson’s conjecture on $\nabla_{p_1}^n$.
                          2. Describe $H^*(\text{Hilb}(x^{nd}=y^n))$ as a module over rational Double Affine Hecke algebra (DAHA) and relate it to the above.


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                              Problem 9.7.

                              Compare recursions for $q,t$-Catalan numbers $C_n(q,t)$ with the recursions for $HHH$.

                                  Cite this as: AimPL: Algebra, geometry, and combinatorics of link homology, available at http://aimpl.org/agclinkhom.