Timeline for Partitions corresponding to unipotent elements in simple classical algebraic groups
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 30, 2019 at 14:14 | comment | added | spin | I think that in that case the partition is $(2,3^2)$. In general for the Levi $A_{n-1}$ in $C_n$, the regular unipotent of $A_{n-1}$ has partition $(n,n)$ on the natural module of $C_n$. The embedding $SL_n < Sp_{2n}$ is given by a maximal totally singular decomposition. | |
| Aug 30, 2019 at 14:05 | review | Close votes | |||
| Sep 1, 2019 at 18:07 | |||||
| Aug 30, 2019 at 13:40 | history | edited | ChockaBlock | CC BY-SA 4.0 |
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| Aug 30, 2019 at 13:37 | comment | added | ChockaBlock | Thanks for the suggestion! I actually find the other types more confusing and am mostly okay with type $A$. One example that's causing me some grief is $\Psi = A_2C_1 < C_4$, where it seems like the partition should be $(3,2,1^3)$ but odd parts need even multiplicity. | |
| Aug 30, 2019 at 13:02 | comment | added | spin | In any case, to find the corresponding partitions, try to identify the subsystem subgroup in terms of the natural module. For example in type A (say $SL_n$) you get subsystem subgroups which look like $SL_{n_1} \times \cdots \times SL_{n_t}$, so the corresponding partition is $(n_1, \ldots, n_t)$ plus some number of $1$'s. | |
| Aug 30, 2019 at 12:58 | comment | added | spin | It is still not completely correct. In type $D$ you have some pairs of unipotent classes corresponding to the same partition. Some unipotent classes of $O_{2n}$ split into two unipotent classes in $SO_{2n}$. | |
| Aug 30, 2019 at 12:33 | comment | added | ChockaBlock | Yes, you're right. I've updated the question to reflect this. | |
| Aug 30, 2019 at 12:33 | history | edited | ChockaBlock | CC BY-SA 4.0 |
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| Aug 30, 2019 at 12:28 | comment | added | spin | Your statement about partitions is only correct when $G$ is of type $A$. For example, for a unipotent element of $Sp_{2n}$, each Jordan block of odd size must have even multiplicity. So for example in $Sp_4$ there is no unipotent element with corresponding partition $(1, 3)$. | |
| Aug 30, 2019 at 12:23 | history | asked | ChockaBlock | CC BY-SA 4.0 |