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Journal article
Cyclage, catabolism, and the affine Hecke algebra
Advances in mathematics (New York. 1965), v 228(4), pp 2292-2351
10 Nov 2011
Abstract
We identify a subalgebra
H
ˆ
n
+
of the extended affine Hecke algebra
H
ˆ
n
of type
A. The subalgebra
H
ˆ
n
+
is a
u-analogue of the monoid algebra of
S
n
⋉
Z
⩾
0
n
and inherits a canonical basis from that of
H
ˆ
n
. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod
n, which we term
positive affine tableaux (PAT).
We then exhibit a cellular subquotient
R
1
n
of
H
ˆ
n
+
that is a
u-analogue of the ring of coinvariants
C
[
y
1
,
…
,
y
n
]
/
(
e
1
,
…
,
e
n
)
with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element
π
∈
H
ˆ
n
+
corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that
R
1
n
has cellular quotients
R
λ
that are
u-analogues of the Garsia–Procesi modules
R
λ
with left cells labeled by (a PAT version of) the
λ-catabolizable tableaux.
We give a conjectural description of a cellular filtration of
H
ˆ
n
+
, the subquotients of which are isomorphic to dual versions of
R
λ
under the perfect pairing on
R
1
n
. This turns out to be closely related to the combinatorics of the cells of
H
ˆ
n
worked out by Shi, Lusztig, and Xi, and we state explicit conjectures along these lines. We also conjecture that the
k-atoms of Lapointe, Lascoux and Morse (2003)
[9] and the
R-catabolizable tableaux of Shimozono and Weyman (2000)
[20] have cellular counterparts in
H
ˆ
n
+
. We extend the idea of atom copies from Lapointe, Lascoux and Morse (2003)
[9] to positive affine tableaux and give descriptions, mostly conjectural, of some of these copies in terms of catabolizability.
Metrics
Details
- Title
- Cyclage, catabolism, and the affine Hecke algebra
- Creators
- Jonah Blasiak - University of Chicago
- Publication Details
- Advances in mathematics (New York. 1965), v 228(4), pp 2292-2351
- Publisher
- Elsevier
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000295193600015
- Scopus ID
- 2-s2.0-80052762814
- Other Identifier
- 991021862256804721
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- Web of Science research areas
- Mathematics