Abstract
We construct complexes \(P_{1^{n}}\) of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture (Flag Hilbert schemes, colored projectors and Khovanov–Rozansky homology. arXiv:1608.07308, 2016) of Gorsky–Neguț–Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for \(P_{1^{n}}\) and its twisted variants. We also show that this homology is also a certain limit of Khovanov–Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest.
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Abbreviations
- (k):
-
Grading shift by k (§2.1)
- \(\alpha _i\) :
-
\(x_i - x_{i+1} = q_{i,i+1}\) (§4.4)
- \({\text {Cone}}\) :
-
Mapping cone (§2.2)
- \({\text {deg}}_h\) :
-
Homological degree (§2.6)
- \(\ell \) :
-
\(\frac{1}{2}n(n-1)\) (§2)
- \(\underline{{\text {End}}}(M)\) :
-
Graded endomorphism ring of M (§2.2)
- \({\text {FT}}_n\) :
-
Rouquier complex of the full-twist braid \({\text {HT}}_n{\text {HT}}_n\) (§3.3)
- \({\text {gr}}\) :
-
Associated graded (§2.6)
- \(\mathcal {H}_n\) :
-
Hecke algebra of \(S_n\) (§2.3)
- \({\text {HH}}(M)\) :
-
\(\bigoplus _{k=0}^\infty {\text {HH}}^k(R;M)\) (§4.1)
- \({\text {HH}}^k(R;M)\) :
-
kth Hochschild cohomology of the (R, R)-bimodule M (§4.1)
- \({\text {HHH}}\) :
-
\(H \circ {\text {HH}}\) (§4.1)
- \({\text {HHH}}^0\) :
-
\(H \circ {\text {HH}}^0 = H\circ {\text {Hom}}\) (§4.1)
- \({\text {hocolim}}A_k\) :
-
Homotopy colimit of the direct system \(\{A_k,f_k\}\) (§3.3)
- \(\underline{{\text {Hom}}}(C,D)\) :
-
Bigraded chain complex of bihomogeneous (R, R)-chain maps (§4.3)
- \(\underline{{\text {Hom}}}(M,N)\) :
-
Graded space of homogeneous (R, R)-bimodule morphisms (§4.3)
- \({\text {HT}}_n\) :
-
Rouquier complex of the half-twist braid \(F_{w_0}\) (§3.3)
- \({\text {HT}}_{min}\) :
-
Minimal complex of \({\text {HT}}_n\) (§3.3)
- \(\mathcal {I}\) :
-
Full subcategory of \(\mathcal {K}^-(\mathbb {S}\mathbf {Bim}_n)\) of chain complexes whose chain bimodules are grading shifts of direct sums of \(B_{w_0}\) (§2.2)
- \(\mathcal {I}^\bot \) :
-
All objects \(C \in \mathcal {K}^-(\mathbb {S}\mathbf {Bim}_n)\) such that \(CB_{w_0} \simeq 0\) (§2.2)
- \(\mathcal {J}_k\) :
-
Full subcategory of \(\mathcal {K}^-(\mathbb {S}\mathbf {Bim}_{n})\) of chain complexes whose chain bimodules are shifts of direct sums of \(R\otimes _{R^{(k)}}R\) (§2.5)
- \(\mathcal {K}(\mathcal {A})\) :
-
Homotopy category of the additive category \(\mathcal {A}\) (§2)
- \(\langle k \rangle \) :
-
Homological degree shift by k (§2.1)
- \(\mathcal {P}(M)\) :
-
Poincaré series of M (§4.4)
- \(\partial _i\) :
-
Divided difference operator \(\partial _i(f) = (f-s(f))/\alpha _i\) (§5.1)
- \(\partial _{1,2,\,\ldots \,,n-1}\) :
-
\(\partial _1 \circ \cdots \circ \partial _{n-1}\) (§5.1)
- \(\phi _b, \psi _b\) :
-
Special quasi-isomorphisms defined in Definition 3.11 (§3.2)
- \(P_{1^{n}}\) :
-
A resolution of R by free graded \(R\otimes _{R^W}R\)-modules (§2.3)
- \(P_{1^{n}}^\vee \) :
-
The dual complex of \(P_{1^{n}}\) (§2.6)
- \(\mathbb {S}\mathbf {Bim}_n\) :
-
Category of Soergel bimodules (§2.1)
- \(\sqcup \) :
-
\(\otimes _{\mathbb {Q}}\) (§2.1)
- \(\theta _k\) :
-
An odd variable of degree \(tq^{-2}\) (§2.4)
- \(\tilde{A}_n,\tilde{M}_n\) :
-
Reduced versions of \(A_n\) and \(M_n\) (See Remark 2.48) (§2.6)
- \({\text {Tot}}(P)\) :
-
Convolution of a sequence of chain complexes (§2.5)
- \(\xi _k, \xi _k^\vee \) :
-
Odd variables of degree \(q^{-2i}a\) and \(q^{2i}a^{-1}\) respectively (§4.1)
- \(\mathbf {x}\) :
-
List of variables \(x_1,\ldots ,x_n\) (§2.1)
- \(\mathbf {y}\) :
-
List of variables \(y_1,\ldots ,y_n\) (§2.1)
- \(^\bot \mathcal {I}\) :
-
All objects \(C \in \mathcal {K}^-(\mathbb {S}\mathbf {Bim}_n)\) such that \(B_{w_0}C \simeq 0\); equivalent to \(\mathcal {I}^\bot \) (§2.2)
- \(A_n\) :
-
Superpolynomial dg-algebra defined in Definition 2.44 (§2.6)
- \(a_{ij}\) :
-
Special polynomials defined in Definition 2.40 (§2.5)
- AB :
-
\(A \otimes _R B\) (§2.1)
- \(B_s\) :
-
Soergel bimodule \(R \otimes _{R^s} R (-1)\) (§2.1)
- \(B_{w_0}\) :
-
Soergel bimodule \(R \otimes _{R^{W} }R(-\ell )\) (§2.1)
- \(B_{w_1}\) :
-
\(R \otimes _{R^{S_{n-1}}} R (-\ell + n -1)\) (§2.1)
- C(i, j):
-
C shifted up by i in q-degree and j in homological degree (§4.1)
- \(d_A\) :
-
Differential on \(A_n\) (see Definition 2.44) (§2.6)
- \(d_M\) :
-
Differential on \(A_n\) (see Definition 2.46) (§2.6)
- E :
-
The algebra defined in Definition 4.19 (§4.3)
- \(e_k\) :
-
k-th elementary symmetric polynomial (§2.1)
- \(F(\beta )\) :
-
Alternate notation for \(F_b\) (§3.3)
- \(F(\beta )_{min}\) :
-
Minimal complex of \(F(\beta )\) (§3.3)
- \(F_b\) :
-
Rouquier complex of the braid \(b\in {\text {Br}}_n\) (§3.2)
- \(F_s^{\pm }\) :
-
Rouquier complex of braid generators \(\sigma _s^{\pm 1}\) (§3.2)
- \(F_{\underline{w}}^\pm \) :
-
Rouquier complex of the positive/negative braid lift of \(w\in S_n\) (§3.2)
- H :
-
Homology functor (§2.2)
- \(h_k\) :
-
kth complete symmetric function (§5.1)
- \(I_k\) :
-
Ideal in \(\mathbb {Q}[\mathbf {x},\mathbf {y}]\) generated by \(e_i(\mathbf {x})-e_i(\mathbf {y}) (1 \le i \le k)\) and \(y_i - x_i (k+1 \le i \le n)\) (§2.1)
- \(J_w\) :
-
Ideal generated by differences \(x_{w(i)}-x_i\) for \(1 \le i \le n\) (§4.4)
- \(M_n\) :
-
dg-\(A_n\)-module defined in Definition 2.46 (§2.6)
- n :
-
A fixed positive integer (§2.1)
- \(p_T\) :
-
Young symmetrizer in \(\mathcal {H}_n\) corresponding to the standard Young tableau T (§2.3)
- \(p_{ij}\) :
-
\(y_i-x_j\) (§2.6)
- \(q_{ij}\) :
-
\(x_i-x_j\) (§4.2)
- R :
-
Polynomial ring \(\mathbb {Q}[x_1,\ldots ,x_n]\) (§2.1)
- \(R^I\) :
-
\(R^{S_{n-1}}\) (§5.1)
- \(R^s\) :
-
Polynomials in R fixed by s (§2.1)
- \(R^W\) :
-
Ring of symmetric polynomials (§2.1)
- \(R^{S_{n-1}}\) :
-
Ring of polynomials symmetric in the first \(n-1\) variables (§2.1)
- \(R_w\) :
-
Standard bimodule associated to \(w \in S_n\) (§3.1)
- s :
-
Simple transposition in \(S_n\) (§2.1)
- \(u_k\) :
-
Even variables of degree \(t^2q^{-2k}\) (§2.6)
- \(v_{ij}\) :
-
Formal even variables of degree \(q^2\) for \(1 \le i < j \le n\) (§4.3)
- W :
-
Symmetric group \(S_n\) (§2.1)
- \(w(P_{1^{n}})\) :
-
Twisted projector (§3.1)
- w(C):
-
\(C \otimes _R R_{w^{-1}}\) (§3.1)
- \(w_0\) :
-
Longest word of \(S_n\) (§2.1)
- \(w_1\) :
-
Longest word of \(S_{n-1}\subset S_n\) (§2.1)
- \(z_{m,n}\) :
-
Special polynomials defined in Proposition 5.11 (§5.2)
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Abel, M., Hogancamp, M. Categorified Young symmetrizers and stable homology of torus links II. Sel. Math. New Ser. 23, 1739–1801 (2017). https://doi.org/10.1007/s00029-017-0336-4
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DOI: https://doi.org/10.1007/s00029-017-0336-4