Affine Nil-Hecke algebras
Eirini Chavli - Tuong-Huy Nguyen
Universit´
e Paris Diderot - Universit´
e Montpellier 2
17 June 2014
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
1 / 14
The symmetric group
Let n ≥ 1.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
2 / 14
The symmetric group
Let n ≥ 1.
We symbolise by Sn the symmetric group on a finite set of n symbols.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
2 / 14
The symmetric group
Let n ≥ 1.
We symbolise by Sn the symmetric group on a finite set of n symbols.
Given i ∈ {1, ....n − 1}, we put si = (i, i + 1) ∈ Sn .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
2 / 14
The symmetric group
Let n ≥ 1.
We symbolise by Sn the symmetric group on a finite set of n symbols.
Given i ∈ {1, ....n − 1}, we put si = (i, i + 1) ∈ Sn .
Proposition
The group Sn admits a presentation with generators
s1 , ..., sn−1
and relations
si2 = 1,
si si+1 si = si+1 si si+1 ,
si sj = sj si , for |i − j| > 1
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
2 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Definition
The minimal integer k such that there is an expression of the above form
is called the length of w and we symbolise it by `(w ).
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Definition
The minimal integer k such that there is an expression of the above form
is called the length of w and we symbolise it by `(w ). Such an expression
is called a reduced decomposition of w .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Definition
The minimal integer k such that there is an expression of the above form
is called the length of w and we symbolise it by `(w ). Such an expression
is called a reduced decomposition of w .
Examples:
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Definition
The minimal integer k such that there is an expression of the above form
is called the length of w and we symbolise it by `(w ). Such an expression
is called a reduced decomposition of w .
Examples:
Let w = s1 s3 s2 s1 s2 ∈ S4 .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Definition
The minimal integer k such that there is an expression of the above form
is called the length of w and we symbolise it by `(w ). Such an expression
is called a reduced decomposition of w .
Examples:
Let w = s1 s3 s2 s1 s2 ∈ S4 .
Then `(w ) = 3, since w = s3 s12 s2 s1 = s3 s2 s1 .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Definition
The minimal integer k such that there is an expression of the above form
is called the length of w and we symbolise it by `(w ). Such an expression
is called a reduced decomposition of w .
Examples:
Let w = s1 s3 s2 s1 s2 ∈ S4 .
Then `(w ) = 3, since w = s3 s12 s2 s1 = s3 s2 s1 .
Let w = s1 s3 . Then `(w ) = 2 and a reduced decomposition is s1 s3 .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Definition
The minimal integer k such that there is an expression of the above form
is called the length of w and we symbolise it by `(w ). Such an expression
is called a reduced decomposition of w .
Examples:
Let w = s1 s3 s2 s1 s2 ∈ S4 .
Then `(w ) = 3, since w = s3 s12 s2 s1 = s3 s2 s1 .
Let w = s1 s3 . Then `(w ) = 2 and a reduced decomposition is s1 s3 .
Mazumoto’s Lemma
Let a a reduced decomposition of w . Then all reduced decompositions of
w can be obtained from a by applying si si+1 si = si+1 si si+1 and
si sj = sj si , for |i − j| > 1
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Let w ∈ Sn . Then, w = si1 ...sik , where ij ∈ {1, ..., n − 1}
Definition
The minimal integer k such that there is an expression of the above form
is called the length of w and we symbolise it by `(w ). Such an expression
is called a reduced decomposition of w .
Examples:
Let w = s1 s3 s2 s1 s2 ∈ S4 .
Then `(w ) = 3, since w = s3 s12 s2 s1 = s3 s2 s1 .
Let w = s1 s3 . Then `(w ) = 2 and a reduced decomposition is s1 s3 .
Mazumoto’s Lemma
Let a a reduced decomposition of w . Then all reduced decompositions of
w can be obtained from a by applying si si+1 si = si+1 si si+1 and
si sj = sj si , for |i − j| > 1 (We don’t need si2 = 1).
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
3 / 14
The symmetric group
Some remarks:
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
4 / 14
The symmetric group
Some remarks:
`(w ) = `(w −1 )
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
4 / 14
The symmetric group
Some remarks:
`(w ) = `(w −1 )
`(w ) = 1 ⇔ w ∈ {s1 , ...sn−1 }
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
4 / 14
The symmetric group
Some remarks:
`(w ) = `(w −1 )
`(w ) = 1 ⇔ w ∈ {s1 , ...sn−1 }
`(w ) = 0 ⇔ w = 1
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
4 / 14
The symmetric group
Some remarks:
`(w ) = `(w −1 )
`(w ) = 1 ⇔ w ∈ {s1 , ...sn−1 }
`(w ) = 0 ⇔ w = 1
Proposition
Let Rw = {(i, j) | i < j and w (i) > w (j)}. Then, `(w ) = |Rw |.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
4 / 14
The symmetric group
Some remarks:
`(w ) = `(w −1 )
`(w ) = 1 ⇔ w ∈ {s1 , ...sn−1 }
`(w ) = 0 ⇔ w = 1
Proposition
Let Rw = {(i, j) | i < j and w (i) > w (j)}. Then, `(w ) = |Rw |.
The element w0 := (1, n)(2, n − 1)(3, n − 2)... is the unique element
.
of Sn with maximal length `(w0 ) = n(n−1)
2
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
4 / 14
The Hecke algebra of the symmetric group
Let R = Z[u1 , u2 ].
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
5 / 14
The Hecke algebra of the symmetric group
Let R = Z[u1 , u2 ].
Definition
The Hecke algebra Hnf of Sn is the R-algebra with generators
T1 , ..., Tn−1
and relations
(Ti − u1 )(Ti − u2 ) = 0,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 ,
Ti Tj = Tj Ti , for |i − j| > 1
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
5 / 14
The Hecke algebra of the symmetric group
Let R = Z[u1 , u2 ].
Definition
The Hecke algebra Hnf of Sn is the R-algebra with generators
T1 , ..., Tn−1
and relations
(Ti − u1 )(Ti − u2 ) = 0,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 ,
Ti Tj = Tj Ti , for |i − j| > 1
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
5 / 14
The Hecke algebra of the symmetric group
Let w ∈ Sn with a reduced decomposition w = si1 ...sik .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
6 / 14
The Hecke algebra of the symmetric group
Let w ∈ Sn with a reduced decomposition w = si1 ...sik .
We put Tw = Ti1 ...Tik ∈ Hnf , which is independent of the choice of a
reduced decomposition of w , by Matsumoto’s lemma.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
6 / 14
The Hecke algebra of the symmetric group
Let w ∈ Sn with a reduced decomposition w = si1 ...sik .
We put Tw = Ti1 ...Tik ∈ Hnf , which is independent of the choice of a
reduced decomposition of w , by Matsumoto’s lemma.
Theorem (Iwahori)
The Hecke algebra of Sn is a free R-module with basis {Tw }w ∈Sn .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
6 / 14
The Hecke algebra of the symmetric group
Let w ∈ Sn with a reduced decomposition w = si1 ...sik .
We put Tw = Ti1 ...Tik ∈ Hnf , which is independent of the choice of a
reduced decomposition of w , by Matsumoto’s lemma.
Theorem (Iwahori)
The Hecke algebra of Sn is a free R-module with basis {Tw }w ∈Sn .
Remarks:
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
6 / 14
The Hecke algebra of the symmetric group
Let w ∈ Sn with a reduced decomposition w = si1 ...sik .
We put Tw = Ti1 ...Tik ∈ Hnf , which is independent of the choice of a
reduced decomposition of w , by Matsumoto’s lemma.
Theorem (Iwahori)
The Hecke algebra of Sn is a free R-module with basis {Tw }w ∈Sn .
Remarks:
Under the specialization u1 = 1, u2 = −1, we take the group algebra
Z[Sn ], where the element Tw becomes the group element w .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
6 / 14
The Hecke algebra of the symmetric group
Let w ∈ Sn with a reduced decomposition w = si1 ...sik .
We put Tw = Ti1 ...Tik ∈ Hnf , which is independent of the choice of a
reduced decomposition of w , by Matsumoto’s lemma.
Theorem (Iwahori)
The Hecke algebra of Sn is a free R-module with basis {Tw }w ∈Sn .
Remarks:
Under the specialization u1 = 1, u2 = −1, we take the group algebra
Z[Sn ], where the element Tw becomes the group element w .
Given w , w 0 ∈ Sn with `(ww 0 ) = `(w ) + `(w 0 ) we have
Tw Tw 0 = Tww 0 .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
6 / 14
The nil Hecke algebra of type A
Definition
The nil-Hecke algebra 0Hnf of type A is the Z-algebra with generators
T1 , ..., Tn−1
and relations
Ti2 = 0,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 ,
Ti Tj = Tj Ti , for |i − j| > 1
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
7 / 14
The nil Hecke algebra of type A
Definition
The nil-Hecke algebra 0Hnf of type A is the Z-algebra with generators
T1 , ..., Tn−1
and relations
Ti2 = 0,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 ,
Ti Tj = Tj Ti , for |i − j| > 1
Remark:
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
7 / 14
The nil Hecke algebra of type A
Definition
The nil-Hecke algebra 0Hnf of type A is the Z-algebra with generators
T1 , ..., Tn−1
and relations
Ti2 = 0,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 ,
Ti Tj = Tj Ti , for |i − j| > 1
Remark:
The 0Hnf algebra is the algebra Hnf under the specialization u1 = u2 = 0.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
7 / 14
The nil-Hecke algebra of type A
Given w , w 0 ∈ Sn we have:
(
Tww 0
Tw Tw 0 =
0
if `(ww 0 ) = `(w ) + `(w 0 )
otherwise.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
8 / 14
The nil-Hecke algebra of type A
Given w , w 0 ∈ Sn we have:
(
Tww 0
Tw Tw 0 =
0
if `(ww 0 ) = `(w ) + `(w 0 )
otherwise.
So, the algebra 0Hnf is graded, if we fix deg Ti = −2.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
8 / 14
The nil-Hecke algebra of type A
Given w , w 0 ∈ Sn we have:
(
Tww 0
Tw Tw 0 =
0
if `(ww 0 ) = `(w ) + `(w 0 )
otherwise.
So, the algebra 0Hnf is graded, if we fix deg Ti = −2.
Or we can say that deg Tw = −2`(w ).
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
8 / 14
The nil-Hecke algebra of type A
Given w , w 0 ∈ Sn we have:
(
Tww 0
Tw Tw 0 =
0
if `(ww 0 ) = `(w ) + `(w 0 )
otherwise.
So, the algebra 0Hnf is graded, if we fix deg Ti = −2.
Or we can say that deg Tw = −2`(w ).
We have 0Hnf = ⊕i∈Z (0Hnf )i , where
(0Hnf )i = ⊕w ∈Sn ,`(w )=−i/2 ZTw .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
8 / 14
The nil-Hecke algebra of type A
Given w , w 0 ∈ Sn we have:
(
Tww 0
Tw Tw 0 =
0
if `(ww 0 ) = `(w ) + `(w 0 )
otherwise.
So, the algebra 0Hnf is graded, if we fix deg Ti = −2.
Or we can say that deg Tw = −2`(w ).
We have 0Hnf = ⊕i∈Z (0Hnf )i , where
(0Hnf )i = ⊕w ∈Sn ,`(w )=−i/2 ZTw .
So, (0Hnf )i = 0 unless i ∈ {0, −2, ..., −n(n − 1)}.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
8 / 14
The nil-Hecke algebra of type A
Given w , w 0 ∈ Sn we have:
(
Tww 0
Tw Tw 0 =
0
if `(ww 0 ) = `(w ) + `(w 0 )
otherwise.
So, the algebra 0Hnf is graded, if we fix deg Ti = −2.
Or we can say that deg Tw = −2`(w ).
We have 0Hnf = ⊕i∈Z (0Hnf )i , where
(0Hnf )i = ⊕w ∈Sn ,`(w )=−i/2 ZTw .
So, (0Hnf )i = 0 unless i ∈ {0, −2, ..., −n(n − 1)}.
Example: The case of 0H3f .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
8 / 14
The nil-Hecke algebra of type A
Given w , w 0 ∈ Sn we have:
(
Tww 0
Tw Tw 0 =
0
if `(ww 0 ) = `(w ) + `(w 0 )
otherwise.
So, the algebra 0Hnf is graded, if we fix deg Ti = −2.
Or we can say that deg Tw = −2`(w ).
We have 0Hnf = ⊕i∈Z (0Hnf )i , where
(0Hnf )i = ⊕w ∈Sn ,`(w )=−i/2 ZTw .
So, (0Hnf )i = 0 unless i ∈ {0, −2, ..., −n(n − 1)}.
Example: The case of 0H3f .
See blackboard!
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
8 / 14
The affine nil-Hecke algebra of type A
Definition
The affine nil Hecke algebra 0Hn of type A is the Z-algebra with generators
T1 , ..., Tn−1
and X1 , . . . , Xn
and relations
Ti2 = 0,
Ti Tj = Tj Ti for |i − j| > 1,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
9 / 14
The affine nil-Hecke algebra of type A
Definition
The affine nil Hecke algebra 0Hn of type A is the Z-algebra with generators
T1 , ..., Tn−1
and X1 , . . . , Xn
and relations
Ti2 = 0,
Ti Tj = Tj Ti for |i − j| > 1,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 .
Xi Xj = Xj Xi ,
Ti Xj = Xj Ti if j − i 6= 0, 1,
Ti Xi+1 − Xi Ti = 1 Ti Xi − Xi+1 Ti = −1.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
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The affine nil-Hecke algebra of type A
Definition
The affine nil Hecke algebra 0Hn of type A is the Z-algebra with generators
T1 , ..., Tn−1
and X1 , . . . , Xn
and relations
Ti2 = 0,
Ti Tj = Tj Ti for |i − j| > 1,
Ti Ti+1 Ti = Ti+1 Ti Ti+1 .
Xi Xj = Xj Xi ,
Ti Xj = Xj Ti if j − i 6= 0, 1,
Ti Xi+1 − Xi Ti = 1 Ti Xi − Xi+1 Ti = −1.
It is a graded algebra with deg Ti = −2 and deg Xi = 2.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
9 / 14
Demazure operators
Let Pn = Z[X1 , ..., Xn ] and we let Sn act on Pn by permutation of the Xi0 s.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
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Affine- Nil-Hecke
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17 June 2014
10 / 14
Demazure operators
Let Pn = Z[X1 , ..., Xn ] and we let Sn act on Pn by permutation of the Xi0 s.
We define an endomorphism of abelian groups ∂i ∈ EndZ (Pn ) by
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
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ealgebras
Montpellier 2)
17 June 2014
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Demazure operators
Let Pn = Z[X1 , ..., Xn ] and we let Sn act on Pn by permutation of the Xi0 s.
We define an endomorphism of abelian groups ∂i ∈ EndZ (Pn ) by
∂i (P) =
P − si (P)
.
Xi+1 − Xi
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
10 / 14
Demazure operators
Let Pn = Z[X1 , ..., Xn ] and we let Sn act on Pn by permutation of the Xi0 s.
We define an endomorphism of abelian groups ∂i ∈ EndZ (Pn ) by
∂i (P) =
P − si (P)
.
Xi+1 − Xi
∂i is well-defined and is PnSn -linear.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
10 / 14
Demazure operators
Let Pn = Z[X1 , ..., Xn ] and we let Sn act on Pn by permutation of the Xi0 s.
We define an endomorphism of abelian groups ∂i ∈ EndZ (Pn ) by
∂i (P) =
P − si (P)
.
Xi+1 − Xi
∂i is well-defined and is PnSn -linear.
Lemma
We have
∂i2 = 0, ∂i ∂i+1 ∂i = ∂i+1 ∂i ∂i+1 and ∂i ∂j = ∂j ∂i , for |i − j| > 1.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
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Affine- Nil-Hecke
Universit´
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Montpellier 2)
17 June 2014
10 / 14
Representation of 0Hnf and 0Hn
Proposition
We obtain a representation of the nil Hecke algebra 0Hnf on Pn with the
assignment
Ti · P = ∂i (P).
Given w ∈ Sn , denote ∂w the image of Tw in EndZ (Pn ).
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
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Montpellier 2)
17 June 2014
11 / 14
Representation of 0Hnf and 0Hn
Proposition
We obtain a representation of the nil Hecke algebra 0Hnf on Pn with the
assignment
Ti · P = ∂i (P).
Given w ∈ Sn , denote ∂w the image of Tw in EndZ (Pn ).
Proposition
We obtain a representation ρ of the affine nil Hecke algebra 0Hn on Pn
with the assignment
ρ(Ti )(P) = ∂i (P),
ρ(Xi )(P) = Xi P.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
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Affine- Nil-Hecke
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17 June 2014
11 / 14
Representation of 0Hnf and 0Hn
Proposition
We obtain a representation of the nil Hecke algebra 0Hnf on Pn with the
assignment
Ti · P = ∂i (P).
Given w ∈ Sn , denote ∂w the image of Tw in EndZ (Pn ).
Proposition
We obtain a representation ρ of the affine nil Hecke algebra 0Hn on Pn
with the assignment
ρ(Ti )(P) = ∂i (P),
ρ(Xi )(P) = Xi P.
Remark : This representation is compatible with the gradings. If
hk ∈ (0 Hn )k then
ρ(hk )((Pn )i ) ⊂ (Pn )i+k .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
11 / 14
Representation of 0Hn
Proposition
This representation ρ of 0Hn on Pn is faithful and we have a decomposition
as Z-modules :
0
Hn = Pn ⊗ 0Hnf .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
12 / 14
Representation of 0Hn
Proposition
This representation ρ of 0Hn on Pn is faithful and we have a decomposition
as Z-modules :
0
Hn = Pn ⊗ 0Hnf .
P
Let a := w Pw Tw ∈ 0Hn (a 6= 0), with {Pw }w ∈Sn a family in Pn .
We will find an element h of 0Hn s.t. ρ(a)(h) 6= 0.
1
Take w1 of minimal length s.t. Pw1 6= 0, then by nice properties of
the longest element w0 :
!
X
a Tw1 −1 w0 =
Pw Tw Tw1 −1 w0 = Pw1 Tw0 .
w
2
Compute the action of this element on h := X2 X32 · · · Xnn−1 , we get
ρ(a)(∂w1 −1 w0 (X2 X32 · · · Xnn−1 )) = Pw1 6= 0.
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
12 / 14
Description of 0Hn as a matrix ring over Pn Sn
Theorem (Bernstein, Zelevinsky, Lusztig)
The center of the affine nil Hecke algebra is the ring of symmetric
polynomials in X1 , . . . , Xn : Z (0Hn ) = Pn Sn .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
13 / 14
Description of 0Hn as a matrix ring over Pn Sn
Theorem (Bernstein, Zelevinsky, Lusztig)
The center of the affine nil Hecke algebra is the ring of symmetric
polynomials in X1 , . . . , Xn : Z (0Hn ) = Pn Sn .
General fact : from the morphism ρ : 0Hn −→ EndZ (Pn ) we get a
morphism of Z (0Hn )-algebras:
0
Hn −→ EndZ (0Hn ) (Pn ).
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
13 / 14
Description of 0Hn as a matrix ring over Pn Sn
Theorem (Bernstein, Zelevinsky, Lusztig)
The center of the affine nil Hecke algebra is the ring of symmetric
polynomials in X1 , . . . , Xn : Z (0Hn ) = Pn Sn .
General fact : from the morphism ρ : 0Hn −→ EndZ (Pn ) we get a
morphism of Z (0Hn )-algebras:
0
Hn −→ EndZ (0Hn ) (Pn ).
Proposition
The previous morphism is an isomorphism of Pn Sn -algebras :
0
∼
Hn −
→ EndPn Sn (Pn ).
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
13 / 14
Description of 0Hn as a matrix ring over Pn Sn
Proposition
The previous morphism is an isomorphism of Pn Sn -algebras :
0
∼
Hn −
→ EndPn Sn (Pn ).
Sketch of proof for the isomorphism :
1
Pn is a progenerator as a 0Hn -module (f.g. projective 0Hn -module and
0H is direct summand of a multiple of P as a 0H -module),
n
n
n
2
0H
n
−→ EndPn Sn (Pn ) is a split injection of Pn Sn -modules,
3
0H
n
is a free Pn Sn -module of rank (n!)2 .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
14 / 14
Description of 0Hn as a matrix ring over Pn Sn
Proposition
The previous morphism is an isomorphism of Pn Sn -algebras :
0
∼
Hn −
→ EndPn Sn (Pn ).
Sketch of proof for the isomorphism :
1
Pn is a progenerator as a 0Hn -module (f.g. projective 0Hn -module and
0H is direct summand of a multiple of P as a 0H -module),
n
n
n
2
0H
n
−→ EndPn Sn (Pn ) is a split injection of Pn Sn -modules,
3
0H
n
is a free Pn Sn -module of rank (n!)2 .
Description of 0Hn as a matrix ring
Since Pn is a free Pn Sn -module of rank n!, the algebra 0Hn is isomorphic to
a (n! × n!)-matrix algebra over Pn Sn .
Eirini Chavli - Tuong-Huy Nguyen (Universit´
e Paris Diderot
Affine- Nil-Hecke
Universit´
ealgebras
Montpellier 2)
17 June 2014
14 / 14
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