Enriched Schubert Galois groups in the Grassmannian G(4,9)

Abraham Martín del Campo, Frank Sottile, and Robert Williams.

Fibered Schubert problems of Type II (from Section 3.2)

Type II fibrations are not as well understood as those of type I. Consequently, their classification in G(4,9) is not as simple as the classificatiomn for type I. There are 29 Type II fibrations.

Theorem 23 gives three classes of Type II fibrations of Schubert problems in Gr(4,9).
Here, the fibration is over a the Schubert problem ⁴ in G(2,4) with fiber a Schubert problem in G(2,5).
As explained following Theorem 23, only three two Schubert problems in G(2,5) may appear as a fiber, and this is described in the Table below.
Type II fibrations over ⁴=2 on Gr(2,4) from Theorem 23

Fiber ²˙² = 2 ˙³ = 2 ˙⁴ = 3

Galois Group S₂WrS₂ S₂WrS₂ S₃WrS₂

Number 11 4 4
Theorem 24 gives Type II fibrations over ˙⁴=3 in G(2,5) with fiber ⁴=2 in G(2,4). Thisa gives five enriched Schubert problems, each of which has Galois group S₂WrS₃, which has order 48.
Theorem 25 gives two more families of Type II fibrations that are fibered over ⁴=2 in G(2,4) with fiber ²˙²=2 in G(2,5). One family has two members and the other has three, and both have Galois group S₂WrS₂, which has order 8.

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Schubert problems having a Type II fibration over ⁴=2 on Gr(2,4) from Theorem 23 with Galois group S₂WrS₂ of order 8
In the field 'fiber' below, a '1' indicates an empty or zero partition.

These have fiber a version of ˙˙˙
Fiber	Schubert Problem	(4)	(2,2)	(2,1,1)	(1,1,1,1)
˙˙˙		12480	18649	12223	6206
˙˙˙1˙		12551	18437	12440	6120
˙˙˙		12544	18589	12220	6157
˙˙˙1˙		12688	18542	12240	6047
˙˙˙		12531	18623	12324	6055
˙˙˙1˙		12601	18597	12266	6063
1˙˙˙˙		12473	18555	12182	6344
1˙˙˙1˙˙		12571	18463	12386	6134
1˙˙˙˙		12549	18597	12186	6191
1˙˙˙˙		12255	18731	12373	6225
1˙˙˙1˙˙		12458	18587	12374	6167

These have fiber a version of ˙˙˙
Fiber	Schubert Problem	(4)	(2,2)	(2,1,1)	(1,1,1,1)
˙˙˙		12549	18499	12222	6279
˙˙˙1˙		12309	18648	12503	6097
1˙˙˙˙		12405	18663	12391	6129
1˙˙˙1˙˙		12579	18661	12241	6110

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Schubert problem having a Type II fibration over ⁴=2 on Gr(2,4) from Theorem 23 with Galois group S₃WrS₂ of order 72
In the field 'fiber' below, a '1' indicates an empty or zero partition.

These have the same Galois group, S₃WrS₂, which has order 72
Fiber	Schubert Problem	(6)	(4,2)	(3,3)	(3,2,1)	(3,1,1,1)	(2,2,2)	(2,2,1,1)	(2,1,1,1,1)	(1,1,1,1,1,1)
˙˙˙˙		8356	12384	2802	8299	2814	4074	6050	3994	692
˙˙˙1˙˙		8274	12438	2720	8196	2708	4137	6306	4015	677
1˙˙˙˙˙		8236	12482	2815	8249	2746	4138	6202	4045	629
1˙˙˙1˙˙˙		8332	12512	2685	8133	2779	4107	6239	4158	652

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Schubert problem having a Type II fibration over ˙⁴=3 on Gr(2,5) from Theorem 24 with fiber ⁴=2 on Gr(2,4)

These have the same Galois group, S₂WrS₃, which has order 48.
Schubert Problem	(6)	(4,2)	(4,1,1)	(3,3)	(2,2,2)	(2,2,1,1)	(2,1,1,1,1)	(1,1,1,1,1,1)
	8200	6219	6186	8445	7101	9317	3059	980
	8259	6196	6185	8207	7153	9290	3220	995
	8309	6304	6033	8385	7366	9201	2929	1008
	8306	6217	6129	8357	7285	9208	2954	1049
	8358	6104	6175	8303	7109	9396	3031	1019

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Schubert problems having a Type II fibration over ²˙²=2 on Gr(2,5) from Theorem 25 with fiber ⁴=2 on Gr(2,4)

These have the same Galois group, S₂[S₂], which has order 8
These form one sporadic family
Schubert Problem	(4)	(2,2)	(2,1,1)	(1,1,1,1)
	12645	18520	12292	6149
	12377	18634	12328	6280

These have the same Galois group, S₂[S₂], which has order 8
These form another sporadic family
Schubert Problem	(4)	(2,2)	(2,1,1)	(1,1,1,1)
	12464	18623	12399	6080
	12408	18734	12261	6150
	12563	18661	12196	6109

Work of Sottile supported by the National Science Foundation under Grant DMS-1501370.

Last modified: Sun Feb 3 15:12:19 CST 2019

Fiber	²˙² = 2	˙³ = 2	˙⁴ = 3
Galois Group	S₂WrS₂	S₂WrS₂	S₃WrS₂
Number	11	4	4