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Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

Abstract

We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

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Notes

  1. 1.

    The subscript “\(\mathrm {KL}\)” comes from the fact that these cup diagrams can be used to give a diagrammatic description of the Kazhdan-Lusztig basis of a parabolic Hecke module of type D with maximal parabolic of type A, [28].

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Acknowledgements

The authors thank Michael Ehrig and Daniel Tubbenhauer for many helpful comments and interesting discussions and Daniel Juteau and Martina Lanini for raising the question how to make the action of the component group explicit in our construction. Parts of this article appeared in the second author’s PhD thesis under the supervision of the first author.

Author information

Correspondence to Arik Wilbert.

Appendix A. Dimension of cohomology

Appendix A. Dimension of cohomology

We construct a cell partition of the topological Springer fiber \(\mathcal {S}^{2m-k,k}_\mathrm {KL}\) generalizing a construction in [14], see also [22, 36] for similar cell partitions for topological Springer fibers of type A. By counting the cells we will obtain an explicit dimension formula for the cohomology in Proposition 69. The Betti numbers of the two-row Springer fibers were independently computed in [23]. His method uses restrictions of Springer representations. Our approach uses an explicit cell partition and thus sheds some additional light on the geometry of the Springer fiber.

A.1. A cell decomposition compatible with intersections

Given cup diagrams \(\mathbf{a},\mathbf{b}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\), we write \(\mathbf{a}\rightarrow \mathbf{b}\) if one of the following conditions is satisfied:

  • The diagrams \(\mathbf{a}\) and \(\mathbf{b}\) are identical except at four not necessarily consecutive vertices \(\alpha<\beta<\gamma <\delta \in \{1,\ldots ,m\}\), where they differ by one of the following local moves:

    (A.1)
  • The diagrams \(\mathbf{a}\) and \(\mathbf{b}\) are identical except at three not necessarily consecutive vertices \(\alpha<\beta <\gamma \in \{1,\ldots ,m\}\), where they differ by one of the following local moves:

    (A.2)

We use these loval moves to define a partial order on \(\mathbb {B}^{2m-k,k}_\mathrm {KL}\) by setting \(\mathbf{a}\prec \mathbf{b}\) if there exists a finite chain of arrows \(\mathbf{a}\rightarrow \mathbf{c}_1 \rightarrow \cdots \rightarrow \mathbf{c}_r \rightarrow \mathbf{b}\).

Remark 59

The local moves (A.1) and (A.2) defined in an ad hoc manner above have a natural geometric interpretation in the context of perverse sheaves (constructible with respect to the Schubert stratification) on isotropic Grassmannians , see [6] and [15].

Let \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) and let \(i_1<i_2<\cdots <i_{\lfloor \frac{k}{2}\rfloor }\) be the left endpoints of cups in \(\mathbf{a}\).

Definition 60

Given a vertex \(r\in \{1,\ldots ,m\}\), we say that a cup connecting vertices \(i<j\) or a ray connected to vertex i is to the right of r if \(r\le i\). Let \(\sigma (r)\) be the number of marked cups and marked rays to the right of r.

We have the following homeomorphism

$$\begin{aligned} \begin{array}{cccl} \overline{\xi }_\mathbf{a}:&{}S_\mathbf{a}&{}\longrightarrow &{}(\mathbb {S}^2)^{\lfloor \frac{k}{2}\rfloor }\\ &{}(x_1,\ldots ,x_m)&{}\longmapsto &{}(y_1,\ldots ,y_{\lfloor \frac{k}{2}\rfloor }) \end{array} \end{aligned}$$
(A.3)

where \(y_{r}=(-1)^{i_r+\sigma (i_r)}x_{i_r}\) and \(1\le r\le \lfloor \frac{k}{2}\rfloor \).

We define the homeomorphism \(t:\mathbb {S}^2\rightarrow \mathbb {S}^2\) as the restriction of the linear endomorphism \((x,y,z)\mapsto (z,y,x)\) of \(\mathbb {R}^3\) to \(\mathbb {S}^2\subseteq \mathbb {R}^3\). Note that \(t(p)=q\). Consider the involutive homeomorphism

$$\begin{aligned} \begin{array}{cccl} \Phi _\mathbf{a}:&{}(\mathbb {S}^2)^{\lfloor \frac{k}{2}\rfloor }&{}\longrightarrow &{}(\mathbb {S}^2)^{\lfloor \frac{k}{2}\rfloor }\\ &{}(y_1,\ldots ,y_{\lfloor \frac{k}{2}\rfloor })&{}\longmapsto &{}\left( y_1,\ldots ,y_s,t(y_{s+1}),\ldots ,t(y_{\lfloor \frac{k}{2}\rfloor })\right) \end{array} \end{aligned}$$
(A.4)

where s is the number of cups which are not to the right of the second leftmost ray (if \(m=k\), in which case there is no second ray, the map (A.4) is the identity).

In the following, the composition of the maps in (A.3) and (A.4) will play a crucial role.

Lemma 61

The map \(\Psi _\mathbf{a}:S_\mathbf{a}\rightarrow (\mathbb {S}^2)^{\lfloor \frac{k}{2}\rfloor }\) defined as the composition \(\Phi _\mathbf{a}\circ \overline{\xi }_\mathbf{a}\) is a homeomorphism.

Example 62

The preimage of \((p,p,p,p)\in (\mathbb {S}^2)^4\) under \(\Psi _\mathbf{a}\), where

is given by \((p,-p,p,-p,p,p,-p,q,-q,q)\).

In order to define a cell decomposition of \(S_\mathbf{a}\) for \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\), we proceed as in [14, §4.5] and associate a graph \(\Gamma _\mathbf{a}=\left( \mathcal {V}(\Gamma _\mathbf{a}),\mathcal {E}(\Gamma _\mathbf{a})\right) \) with each cup diagram \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) as follows:

  • The vertices \(\mathcal {V}(\Gamma _\mathbf{a})\) of \(\Gamma _\mathbf{a}\) are given by the cups (ij) in \(\mathbf{a}\).

  • Two vertices \((i_1,j_1),(i_2,j_2) \in \mathcal {V}(\Gamma _\mathbf{a})\) are connected by an edge \((i_1,j_1)-(i_2,j_2)\in \mathcal {E}(\Gamma _\mathbf{a})\) in \(\Gamma _\mathbf{a}\) if and only if there exists some \(\mathbf{b}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) with \(\mathbf{b}\rightarrow \mathbf{a}\) such that \(\mathbf{a}\) is obtained from \(\mathbf{b}\) by a local move of type I)-IV) at the vertices \(i_1,i_2,j_1,j_2\).

As in [14], the graph \(\Gamma _\mathbf{a}\) is a forest whose roots \(\mathcal {R}(\Gamma _\mathbf{a})\) are precisely the outer cups of \(\mathbf{a}\), i.e. the cups which are not nested in any other cup and do not contain any marked cup to their right.

We assign to each subset \(J \subseteq \mathcal {R}(\Gamma _\mathbf{a}) \cup \mathcal {E}(\Gamma _\mathbf{a})\) the subset \(C_J'\) of \((\mathbb {S}^2)^{\lfloor \frac{k}{2}\rfloor }\) given by all elements \((y_1,\ldots ,y_{\lfloor \frac{k}{2}\rfloor })\in (\mathbb {S}^2)^{\lfloor \frac{k}{2}\rfloor }\) which satisfy the following relations:

  1. (C1)

    If \((i_q,j_q) \in \mathcal {R}(\Gamma _\mathbf{a}) \cap J\) then \(y_q = p\),

  2. (C2)

    if \((i_q,j_q) \in \mathcal {R}(\Gamma _\mathbf{a})\) but \((i_q,j_q) \not \in J\) then \(y_q \ne p\),

  3. (C3)

    if \((i_q,j_q) - (i_{q'},j_{q'}) \in \mathcal {E}(\Gamma _\mathbf{a}) \cap J\) then \(y_q = y_{q'}\),

  4. (C4)

    if \((i_q,j_q) - (i_{q'},j_{q'}) \not \in \mathcal {E}(\Gamma _\mathbf{a}) \cap J\) then \(y_q \ne y_{q'}\).

Lemma 63

There is a decomposition

$$\begin{aligned} (\mathbb {S}^2)^{\lfloor \frac{k}{2}\rfloor } = \bigsqcup _{J \subseteq \mathcal {R}(\Gamma _\mathbf{a})\cup \mathcal {E}(\Gamma _\mathbf{a})} C_J' \end{aligned}$$
(A.5)

into disjoint cells \(C_J'\) homeomorphic to \(\mathbb {R}^{2(\lfloor \frac{k}{2}\rfloor -|J|)}\). Moreover, pushing forward along (A.3) gives a cell decomposition

$$\begin{aligned} S_\mathbf{a}= \bigsqcup _{J \subseteq \mathcal {R}(\Gamma _\mathbf{a})\cup \mathcal {E}(\Gamma _\mathbf{a})} C_J, \end{aligned}$$
(A.6)

where \(C_J = (\Psi _\mathbf{a})^{-1}(C_J')\).

Proof

The proof of (A.5) is the same as in the equal-row case [14, Lemma 4.17] because the additional rays do not play any role in the construction of the \(C_J'\), and (A.5) implies (A.6). \(\square \)

The reason for choosing the homeomorphism \(\Psi _\mathbf{a}\) in the construction of the cell decomposition (A.6) of \(S_\mathbf{a}\) is that the resulting decompositions are compatible with pairwise intersections in the sense of the next lemma which extends [14, Proposition 4.24] to the general two-block case.

Lemma 64

Let \(\mathbf{a},\mathbf{b}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) such that \(\mathbf{b}\rightarrow \mathbf{a}\).

  1. (1)

    If \(\mathbf{b}\rightarrow \mathbf{a}\) is of type I)-IV), then

    $$\begin{aligned} S_\mathbf{a}\cap S_\mathbf{b}= \bigcup _{J \subseteq \mathcal {R}(\Gamma _\mathbf{a}) \cup \mathcal {E}(\Gamma _\mathbf{a}), e\in J} C_J, \end{aligned}$$

    where \(e \in \mathcal {E}(\Gamma _\mathbf{a})\) is the edge of \(\Gamma _\mathbf{a}\) determined by the move \(\mathbf{b}\rightarrow \mathbf{a}\).

  2. (2)

    If \(\mathbf{b}\rightarrow \mathbf{a}\) is of type I’)-IV’), then there is a unique cup \(\alpha \in \mathrm{cups}(\mathbf{a})\) such that \(\alpha \notin \mathrm{cups}(\mathbf{b})\). Moreover, \(\alpha \in \mathcal {R}(\Gamma _\mathbf{a})\) and

    $$\begin{aligned} S_\mathbf{a}\cap S_\mathbf{b}= \bigcup _{J \subseteq \mathcal {R}(\Gamma _\mathbf{a}) \cup \mathcal {E}(\Gamma _\mathbf{a}), \alpha \in J} C_J. \end{aligned}$$

Proof

This is a straightforward case-by-case analysis. \(\square \)

A.2. A cell partition of the topological Springer fiber

Given a cup diagram \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\), define its extension\(\widetilde{\mathbf{a}}\in \mathbb {B}^{2m-k,2m-k}_\mathrm {KL}\) as the unique cup diagram obtained by connecting the loose endpoints of the \(m-k\) rightmost rays to \(m-k\) newly added vertices to the right of \(\mathbf{a}\) via an unmarked cup such that the resulting diagram is crossingless (see [36, Definition 3.11] and [15, §5.3] for a similar but different extension). Let \(\mathbb {B}^{2m-k,2m-k}_{2m-k,k}\) denote the cup diagrams in \(\mathbb {B}^{2m-k,2m-k}_\mathrm {KL}\) which are obtained as an extension of some diagram in \(\mathbb {B}^{2m-k,k}_\mathrm {KL}\), i.e. the set of all cup diagrams in \(\mathbb {B}^{2m-k,2m-k}_\mathrm {KL}\) whose \(m-k\) rightmost vertices are right endpoints of unmarked cups. If \(m=k\), then the extension procedure returns the same diagram. Note that the leftmost ray is never replaced by a cup.

Example 65

Here is an example showing the extension of a cup diagram:

For illustrative purposes the components of the diagrams which change during extension are drawn in dashed font.

Inspired by the combinatorial completion procedure we define an embedding (see also [36, Section 5] for a similar map in type A)of topological Springer fibers

$$\begin{aligned} \eta _{2m-k,k}:\mathcal {S}^{2m-k,k}_\mathrm {KL}\hookrightarrow \mathcal {S}^{2m-k,2m-k}_\mathrm {KL}\,,\,\,(x_1,\ldots ,x_m)\mapsto (x_1,\ldots ,x_m,-q,\ldots ,-q). \end{aligned}$$

Note that \(\eta _{2m-k,k}(S_\mathbf{a})\subseteq S_{\widetilde{\mathbf{a}}}\).

Given \(\mathbf{a},\mathbf{b}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\), one easily verifies that \(\mathbf{b}\rightarrow \mathbf{a}\) if and only if \(\widetilde{\mathbf{b}}\rightarrow \widetilde{\mathbf{a}}\). Thus, we have an isomorphism of posets \(\mathbb {B}^{2m-k,k}_\mathrm {KL}\cong \widetilde{\mathbb {B}}^{2m-k,2m-k}_\mathrm {KL}\). We equip \(\widetilde{\mathbb {B}}^{2m-k,2m-k}_\mathrm {KL}\) with the induced total order from some fixed total order (which extends the partial order \(\prec \) from before) on \(\mathbb {B}^{2m-k,2m-k}_\mathrm {KL}\). We pull this total order over to \(\mathbb {B}^{2m-k,k}_\mathrm {KL}\) via the isomorphism of posets.

Let \(S_{<\mathbf{a}} = \bigcup _{\mathbf{b}<\mathbf{a}} S_\mathbf{b}\).

Lemma 66

For any \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) we have

$$\begin{aligned} S_{<\mathbf{a}} \cap S_\mathbf{a}= & {} \bigcup _{\mathbf{b}\rightarrow \mathbf{a}} (S_\mathbf{b}\cap S_\mathbf{a}). \end{aligned}$$
(A.7)

Proof

In case \(m=k\) the claimed equality was proven in [14, Lemma 4.23].

Since \(\rightarrow \) implies <, the left side is contained in the right side.

So, assume there exists \(\mathbf{b}<\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) and \(x\in S_\mathbf{a}\cap S_\mathbf{b}\) such that \(x\notin \bigcup _{\mathbf{b}\rightarrow \mathbf{a}} S_\mathbf{b}\cap S_\mathbf{a}\). Then \(\eta _{2m-k,k}(x)\in S_{\widetilde{\mathbf{b}}}\cap S_{\widetilde{\mathbf{a}}}\) (note that \(\widetilde{\mathbf{b}}<\widetilde{\mathbf{a}}\) since \(\mathbf{b}<\mathbf{a}\) and by definition of the total order < on \(\mathbb {B}^{2m-k,k}_\mathrm {KL}\)). Hence,

$$\begin{aligned} \bigcup _{\overset{\mathbf{b}<\widetilde{\mathbf{a}}}{\mathbf{b}\in \mathbb {B}^{2m-k,2m-k}_\mathrm {KL}}} S_\mathbf{b}\cap S_{\widetilde{\mathbf{a}}} = \bigcup _{\overset{\mathbf{b}\rightarrow \widetilde{\mathbf{a}}}{\mathbf{b}\in \mathbb {B}^{2m-k,2m-k}_\mathrm {KL}}} S_\mathbf{b}\cap S_{\widetilde{\mathbf{a}}} = \bigcup _{\overset{\mathbf{b}\rightarrow \widetilde{\mathbf{a}}}{\mathbf{b}\in \widetilde{\mathbb {B}}^{2m-k,2m-k}_\mathrm {KL}}} S_\mathbf{b}\cap S_{\widetilde{\mathbf{a}}} \end{aligned}$$

and therefore

$$\begin{aligned} \eta _{2m-k,k}(\mathcal {S}^{2m-k,k}_\mathrm {KL})\cap \bigcup _{\overset{\mathbf{b}<\widetilde{\mathbf{a}}}{\mathbf{b}\in \mathbb {B}^{2m-k,2m-k}_\mathrm {KL}}} S_\mathbf{b}\cap S_{\widetilde{\mathbf{a}}}&= \eta _{2m-k,k}(\mathcal {S}^{2m-k,k}_{\mathrm {KL}})\cap \bigcup _{\overset{\mathbf{b}\rightarrow \widetilde{\mathbf{a}}}{\mathbf{b}\in \widetilde{\mathbb {B}}^{2m-k,2m-k}_\mathrm {KL}}} S_\mathbf{b}\cap S_{\widetilde{\mathbf{a}}}\\&= \eta _{2m-k,k}\left( \bigcup _{\mathbf{b}\rightarrow \mathbf{a}} S_\mathbf{b}\cap S_\mathbf{a}\right) . \end{aligned}$$

Since \(\eta _{2m-k,k}(x)\) is contained in the leftmost set above, it is also contained in the rightmost set. By the injectivity of \(\eta _{2m-k,k}\), we deduce that \(x\in \bigcup _{b\rightarrow a} S_\mathbf{b}\cap S_\mathbf{a}\), a contradiction. \(\square \)

Fix a cup diagram \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\). A cup in \(\mathbf{a}\) is called special if there exists some \(\mathbf{b}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) such that \(\mathbf{b}\rightarrow \mathbf{a}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are related by a local move of type I’)-IV’), and this cup is the unique cup which changes under the move \(\mathbf{a}\rightarrow \mathbf{b}\). Let \(\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\) be the set of special cups in \(\mathbf{a}\). In particular, we have \(\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}=\emptyset \) if \(k=m\) is even, i.e. if there are no rays in \(\mathbf{a}\).

Remark 67

As in [14, Remark 4.25], the moves I’)-IV’) imply that all outer cups (in the sense as defined above) of a given cup diagram \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) are special if the leftmost ray in \(\mathbf{a}\) is marked, whereas only outer cups to the right of the leftmost ray are special if the ray is unmarked.

Corollary 68

Given \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\), then

$$\begin{aligned} S_{<\mathbf{a}} \cap S_\mathbf{a}= & {} \underset{\begin{array}{c} J \subseteq \mathcal {R}(\Gamma _\mathbf{a}) \cup \mathcal {E}(\Gamma _\mathbf{a}) \\ J \cap (\mathcal {E}(\Gamma _\mathbf{a}) \cup \mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp})\ne \emptyset \end{array}}{\bigcup } C_J. \end{aligned}$$

Proof

This follows directly from the previous two lemmas. \(\square \)

In order to argue inductively in the proof of the following proposition, we introduce the notion of a partial/one-step extension. This is defined as the ordinary extension introduced above, except that we only replace the rightmost ray with a cup.

Proposition 69

If \(k\ne m\), then we have the following dimension formula:

$$\begin{aligned} \dim H^*(\mathcal {S}^{2m-k,k}_\mathrm {KL}) = \sum _{i=0}^{\frac{k-1}{2}}\left( {\begin{array}{c}m\\ i\end{array}}\right) . \end{aligned}$$

Remark 70

In the special case \(m=k\) we have \(\dim H^*(\mathcal {S}^{2m-k,k}_\mathrm {KL})=\sum _{i=0}^{\frac{m-1}{2}}\left( {\begin{array}{c}m\\ i\end{array}}\right) =2^{m-1}\) as proven in [14, Prop. 4.28].

Proof

We have a disjoint union \(\mathcal {S}^{2m-k,k}_\mathrm {KL}= \bigcup S_{<\mathbf{a}} \cap S_\mathbf{a}\). We thus obtain a cell partition of \(\mathcal {S}^{2m-k,k}_\mathrm {KL}\) by using the cell partition from Corollary 68 for each of the \(S_{<\mathbf{a}} \cap S_\mathbf{a}\). Hence, the dimension of \(H^*(\mathcal {S}^{2m-k,k}_\mathrm {KL})\) can be calculated by counting the cells contained in \(S_\mathbf{a}{\setminus } S_{<\mathbf{a}}\) for all \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\) and taking their sum. By Corollary 68 the cells contained in \(S_\mathbf{a}{\setminus } S_{<\mathbf{a}}\) correspond bijectively to the outer cups in \(\mathbf{a}\) which are not special. This yields

$$\begin{aligned} \dim H^*(\mathcal {S}^{2m-k,k}_\mathrm {KL})=\sum _{\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}} 2^{\vert \mathcal {R}(\Gamma _\mathbf{a}){\setminus }\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\vert }. \end{aligned}$$
(A.8)

It remains to show that the right hand side of (A.8) equals \(\sum _{i=0}^{\frac{k-1}{2}}\left( {\begin{array}{c}m\\ i\end{array}}\right) \). We prove this claim by induction on \(m-k\), i.e. by induction on the number of rays in the cup diagrams contained in \(\mathbb {B}^{2m-k,k}_\mathrm {KL}\). Note that the base \(m-k=0\) of the induction (i.e. no ray or one ray; depending on whether m is even or odd) is true by Remark 70.

Thus, we consider a partition \((2m-k,k)\), \(m-k>1\). Firstly, note that we have

$$\begin{aligned} 2^{\vert \mathcal {R}(\Gamma _\mathbf{a}){\setminus }\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\vert }=2^{\vert \mathcal {R}(\Gamma _{\tilde{\mathbf{a}}}){\setminus }\mathcal {R}(\Gamma _{\tilde{\mathbf{a}}})^\mathrm{sp}\vert } \end{aligned}$$
(A.9)

for all \(\mathbf{a}\in \mathbb {B}^{2m-k,k}_\mathrm {KL}\), where \(\tilde{\mathbf{a}}\in \mathbb {B}^{2m-k,k+2}_\mathrm {KL}\) denotes the one-step extension of \(\mathbf{a}\) (note that the completed cup is always special). Secondly, the elements \(\mathbf{a}\in \mathbb {B}^{2m-k,k+2}_\mathrm {KL}{\setminus }\widetilde{\mathbb {B}}^{2m-k,k+2}_\mathrm {KL}\) are precisely the cup diagrams with an unmarked ray connected to the rightmost vertex \(m+1\). Deleting this ray induces a bijection between \(\mathbb {B}^{2m-k,k+2}_\mathrm {KL}{\setminus }\widetilde{\mathbb {B}}^{2m-k,k+2}_\mathrm {KL}\) and \(\mathbb {B}^{2(m-1)-k,k+2}_\mathrm {KL}\) and we have

$$\begin{aligned} 2^{\vert \mathcal {R}(\Gamma _\mathbf{a}){\setminus }\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\vert }=2^{\vert \mathcal {R}(\Gamma _{\mathbf{b}}){\setminus }\mathcal {R}(\Gamma _{\mathbf{b}})^\mathrm{sp}\vert }, \end{aligned}$$
(A.10)

where \(\mathbf{b}\) is the diagram obtained from \(\mathbf{a}\) by deleting the rightmost ray. By using (A.9) in the first and (A.10) in the third equality we compute

$$\begin{aligned} \sum _{a\in \mathbb {B}^{2m-k,k}_\mathrm {KL}} 2^{\vert \mathcal {R}(\Gamma _\mathbf{a}){\setminus }\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\vert }&= \sum _{\tilde{\mathbf{a}}\in \widetilde{\mathbb {B}}^{2m-k,k+2}_\mathrm {KL}} 2^{\vert \mathcal {R}(\Gamma _{\tilde{\mathbf{a}}}){\setminus }\mathcal {R}(\Gamma _{\tilde{\mathbf{a}}})^\mathrm{sp}\vert } \\&= \sum _{\mathbf{a}\in \mathbb {B}^{2m-k,k+2}_\mathrm {KL}} 2^{\vert \mathcal {R}(\Gamma _\mathbf{a}){\setminus }\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\vert } - \sum _{\mathbf{a}\in \mathbb {B}^{2m-k,k+2}_\mathrm {KL}{\setminus }\widetilde{\mathbb {B}}^{2m-k,k+2}_\mathrm {KL}} 2^{\vert \mathcal {R}(\Gamma _\mathbf{a}){\setminus }\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\vert } \\&= \sum _{\mathbf{a}\in \mathbb {B}^{2m-k,k+2}_\mathrm {KL}} 2^{\vert \mathcal {R}(\Gamma _\mathbf{a}){\setminus }\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\vert } - \sum _{\mathbf{a}\in \mathbb {B}^{2(m-1)-k,k+2}_\mathrm {KL}} 2^{\vert \mathcal {R}(\Gamma _\mathbf{a}){\setminus }\mathcal {R}(\Gamma _\mathbf{a})^\mathrm{sp}\vert }. \end{aligned}$$

By (A.8) the two terms in the above difference equal the respective dimension of cohomology which we know by induction. Thus, the above chain of equalities equals

$$\begin{aligned} \sum _{i=0}^{\frac{k+1}{2}}\left( {\begin{array}{c}m+1\\ i\end{array}}\right) - \sum _{i=0}^{\frac{k+1}{2}}\left( {\begin{array}{c}m\\ i\end{array}}\right) =\sum _{i=1}^{\frac{k+1}{2}}\left( \left( {\begin{array}{c}m+1\\ i\end{array}}\right) -\left( {\begin{array}{c}m\\ i\end{array}}\right) \right) = \sum _{i=1}^{\frac{k+1}{2}}\left( {\begin{array}{c}m\\ i-1\end{array}}\right) = \sum _{i=0}^{\frac{k-1}{2}}\left( {\begin{array}{c}m\\ i\end{array}}\right) , \end{aligned}$$

which proves the claim. \(\square \)

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Stroppel, C., Wilbert, A. Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory. Math. Z. 292, 1387–1430 (2019) doi:10.1007/s00209-018-2161-7

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Keywords

  • Springer fiber
  • Action on cohomology
  • Springer theory
  • Diagram algebras
  • Flag varieties
  • Betti numbers

Mathematics Subject Classification

  • Primary 14M15
  • 17B08
  • 17B10
  • Secondary 05E10
  • 20C08
  • 20F36