Tedyun: Created page with "The topology of restricted partition posets - Richard Ehrenborg and JiYoon Jung http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAO0126/3556 Reflection a..."

2012-12-04T01:27:13Z

Created page with "*The topology of restricted partition posets - Richard Ehrenborg and JiYoon Jung http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAO0126/3556 *Reflection a..."

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*The topology of restricted partition posets - Richard Ehrenborg and JiYoon Jung
http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAO0126/3556

*Reflection arrangements and ribbon representations - Alexander Miller
http://arxiv.org/abs/1108.1429

*PARTITIONS INTO EVEN AND ODD BLOCK SIZE AND SOME UNUSUAL CHARACTERS OF THE SYMMETRIC GROUPS - A. R. CALDERBANK, P. HANLON, and R. W. ROBINSON
http://plms.oxfordjournals.org/content/s3-53/2/288.full.pdf
- the top homology group of the order complex of $\Pi_n^d\setminus \{\hat{1}\}$ is the Specht module on the border strip correspoding to the composition $(d, \ldots, d, d-1)$.

*A basis for the d-divisible partition lattice - Wachs
http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6W9F-45NHY22-2W-1&_cdi=6681&_user=501045&_pii=S0001870896900146&_origin=browse&_zone=rslt_list_item&_coverDate=02%2F10%2F1996&_sk=998829997&wchp=dGLzVzb-zSkzk&md5=9761609d478d4cd792d53c224a778067&ie=/sdarticle.pdf
- showed that the d-divisible partition lattice has EL-shelling.

{|
|$\Pi_n^d$ || $\tilde{H}(\Pi_n^d)$ || dimension || Mobius function
|-
|$\Pi^\bullet_{\vec{c}}$ || $\tilde{H}(\Pi^\bullet_{\vec{c}})$ || dimension || Mobius function
|}

== Question ==

# $\tilde{H}_{n-k-1}(\Pi_{(k,1,\ldots,1)}^\bullet \setminus \{\hat{1}\}) \cong_{S_n} \tilde{H}_{n-k-1}(B_n^k \setminus \{\hat{1}\}) \cong_{S_n} \text{Specht module of the border strip } (k,1,\ldots,1).$ Why? <br /> $\Pi_{(k,1,\ldots,1)}^\bullet$ is the set of pointed partitions $\{B_1, B_2, \ldots, B_t, \underline{Z}\}$ such that at least one of $B_i$ has cardinality greater than or equal to $k$. On the other hand, $B_n^k = \{A \in B_n ~\vert ~ \lvert A\rvert \geq k\}$.
#* Answer: $\tilde{H}_{n-k-1}(\Delta(\Pi_{(k,1,\ldots,1)}^\bullet \setminus \{\hat{1}\})) \cong \tilde{H}_{n-k-1} (\Delta_{(k,1,\ldots,1)}\setminus \{\hat{1}\}) \cong \tilde{H}_{n-k-1} (B_n(S)\setminus \{\hat{1}\}) \cong \tilde{H}_{n-k-1}(B_n^k \setminus \{\hat{1}\})$, where $S = \{k, k+1, \ldots, n\}$

# What can we say about the Stanley-Reisner ring of $\Pi_{\vec{c}}^\bullet$?

== Remarks ==

# Wach's EL-labeling of $d$-divisible partition lattice does not work in the case of pointed partition poset because some the rising chain in an interval might be missing in the pointed partition poset.

Restricted Partition Posets and Specht Modules - Revision history

Tedyun: Created page with "*The topology of restricted partition posets - Richard Ehrenborg and JiYoon Jung http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAO0126/3556 *Reflection a..."

Tedyun: Created page with "The topology of restricted partition posets - Richard Ehrenborg and JiYoon Jung http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAO0126/3556 Reflection a..."