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General LLT Polynomials
CRM Workshop on Interactions between Algebraic Combinatorics and Algebraic Geometry
Mark Haiman
joint work with
Ian Grojnowski
May 28, 2007 2
General LLT Polynomials
Part I
The combinatorial polynomials of
Lascoux,Leclerc &Thibon
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General LLT Polynomials
The kcore and kquotient of a partition
(k = 4)
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General LLT Polynomials
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General LLT Polynomials
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General LLT Polynomials
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General LLT Polynomials
The 4quotient ofour partition
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General LLT Polynomials
Removing 4ribbons...
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General LLT Polynomials
Removing 4ribbons...
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General LLT Polynomials
Removing 4ribbons...
May 28, 2007 11
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 12
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 13
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 14
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 15
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 16
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 17
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 18
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 19
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 20
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 21
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 22
General LLT Polynomials
Removing 4ribbons...
May 28, 2007 23
General LLT Polynomials
Removing 4ribbons...
What's left is the 4core
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General LLT Polynomials
Remembering the ordergives a ribbon tableau...
7
6
54
3
21
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General LLT Polynomials
Remembering the ordergives a ribbon tableau...
7
6
54
3
21
...and corresponding standard tableau on the kquotient.
1
23
45
6
7
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General LLT Polynomials
Remembering the ordergives a ribbon tableau...
7
6
54
3
21
...and corresponding standard tableau on the kquotient.
This also works for semistandard tableaux.
1
23
45
6
7
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General LLT Polynomials
Aligning content lines in the kquotient isequivalent to rememberingthe kcore.
1
1
0
012
0
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General LLT Polynomials
Aligning content lines in the kquotient isequivalent to rememberingthe kcore — then wecan use skew shapes.
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General LLT Polynomials
Content reading word of a tableauon a tuple of skew shapes:
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9 38 2 8 8 6 1 8 4 4
61 81 1 4 2 1 6
3 9 6 4 2
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General LLT Polynomials
Inversions in the reading word countif they are on the same line—
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9 38 2 8 8 6 1 8 4 4
61 81 1 4 2 1 6
3 9 6 4 2
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General LLT Polynomials
Inversions in the reading word countif they are on the same line—
10
12
23
21
34
56
1
1
11
1
2
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9 38 2 8 8 6 1 8 4 4
61 81 1 4 2 1 6
3 9 6 4 2
or from aboveto below on
consecutive lines.
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General LLT Polynomials
Inversions in the reading word countif they are on the same line—
10
12
23
21
34
56
1
1
11
1
2
2
2
3
3
4
4
4
4
8
6
6
6
6
8
8
8
8
9
9
9 38 2 8 8 6 1 8 4 4
61 81 1 4 2 1 6
3 9 6 4 2
or from aboveto below on
consecutive lines.
This one does not count!
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General LLT Polynomials
Definition. The spin of a ribbon tableau is the sum overits ribbons
Proposition. If
semistandard ribbon tableau tuple of semistandard tableaux ,
then for some constant depending only on the shape,we have
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General LLT Polynomials
Definition. Given a tuple of skew shapes (with contentalignment), the associated LLT polynomial is thegenerating function for semistandard tableaux
May 28, 2007 35
General LLT Polynomials
Definition. Given a tuple of skew shapes (with contentalignment), the associated LLT polynomial is thegenerating function for semistandard tableaux
[Remark. By the preceding proposition, isthe spin generating function for ribbon tableaux.]
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General LLT Polynomials
Definition. Given a tuple of skew shapes (with contentalignment), the associated LLT polynomial is thegenerating function for semistandard tableaux
[Remark. By the preceding proposition, isthe spin generating function for ribbon tableaux.]
Theorem (L, L & T). is a symmetric function.
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General LLT Polynomials
Theorem (L, L & T). is a symmetric function. Leclerc and Thibon conjectured that is alwaysSchur positive, and proved it for straight shapes. In thiscase, the coefficients in the Schur expansion areparabolic KazhdanLusztig polynomials.
Haglund, H., and Loehr showed that the Macdonaldpolynomials are positive combinations ofLLT polynomials for certain skew shapes.
May 28, 2007 38
General LLT Polynomials
Theorem (L, L & T). is a symmetric function. Leclerc and Thibon conjectured that is alwaysSchur positive, and proved it for straight shapes. In thiscase, the coefficients in the Schur expansion areparabolic KazhdanLusztig polynomials.
Haglund, H., and Loehr showed that the Macdonaldpolynomials are positive combinations ofLLT polynomials for certain skew shapes.
We prove the LeclercThibon conjecture for all shapes.
May 28, 2007 39
General LLT Polynomials
Part II LLT Polynomials in the general case
Notation:
May 28, 2007 40
General LLT Polynomials
Hecke algebra has basisand relations
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General LLT Polynomials
Hecke algebra has basisand relations
Bernstein's presentation of the Affine Hecke algebra:
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General LLT Polynomials
Hecke algebra has basisand relations
Bernstein's presentation of the Affine Hecke algebra:
The center of the affine Hecke algebra is
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General LLT Polynomials
Submodule affords the “trivial” representation
where .
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General LLT Polynomials
Theorem (Lusztig). Let be the maximal element of the double coset . The corresponding KazhdanLusztig basis element in the affine Hecke algebra is
Submodule affords the “trivial” representation
where .
where is an irreducible character of , viewed as an element of the center .
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General LLT Polynomials
Submodule affords the sign representation
of , where
May 28, 2007 46
General LLT Polynomials
Submodule affords the sign representation
of , where
The elements
where is regular and dominant for ,and is a minimal double cosetrepresentative in , form a basis of the space
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General LLT Polynomials
Note that is a module.
The operator of multiplication by on the basishas matrix entries denoted by
May 28, 2007 48
General LLT Polynomials
Note that is a module.
The operator of multiplication by on the basishas matrix entries denoted by
Definition. The generating function of matrix entries
taken over all , for fixed and , is an LLTpolynomial.
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General LLT Polynomials
Note that is a module.
The operator of multiplication by on the basishas matrix entries denoted by
Definition. The generating function of matrix entries
taken over all , for fixed and , is an LLTpolynomial.
[Remark: it's really an infinite formal qcharacter of .]
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General LLT Polynomials
Proposition. Let be such that for all . Then, formally, at we have
In other words, the coefficient of in is equal to the multiplicity of
in .
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General LLT Polynomials
Proposition. Let be such that for all . Then, formally, at we have
In other words, the coefficient of in is equal to the multiplicity of
in .
For , , we get aproduct of skew Schur functions
where and .
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General LLT Polynomials
Proposition. Let be such that for all . Then, formally, at we have
In other words, the coefficient of in is equal to the multiplicity of
in .
Warning: when , the LLT polynomial
depends on the choice of !
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General LLT Polynomials
Theorem. For , , and , if we take , where ,then
May 28, 2007 54
General LLT Polynomials
Theorem. For , , and , if we take , where ,then
Corollary. The coefficients in the expansion throughSchur functions of any LLT polynomial arepositive (i.e., lie in ).
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General LLT Polynomials
Theorem. For , , and , if we take , where ,then
Corollary. The coefficients in the expansion throughSchur functions of any LLT polynomial arepositive (i.e., lie in ).
Corollary. The coefficients in the expansion throughSchur functions of the Macdonald polynomialsare positive.
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General LLT Polynomials
Part III A few words about proofs—
Two things to prove...
May 28, 2007 57
General LLT Polynomials
Part III A few words about proofs—
Two things to prove...First, theorem from previous slide: for ,
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General LLT Polynomials
Part III A few words about proofs—
Two things to prove...First, theorem from previous slide: for ,
We adapt the method of Leclerc and Thibon.
Rename the basis of , denoting basis elementsby , where is a partition with fixed kcore and atmost n parts. When is expressed using the Bernsteingenerators, its definition extends naturally to all ina orbit, with simple straightening relations.
May 28, 2007 59
General LLT Polynomials
Two things to prove...First, theorem from previous slide: for ,
This follows easily from the straightening relations.
Everything but the combinatorial action of worksfor any and .
It's enough to show that the operator acts byadding a vertical ribbon strip to , with coefficient
May 28, 2007 60
General LLT Polynomials
Two things to prove...First, theorem from previous slide: for ,
This follows easily from the straightening relations.
It's enough to show that the operator acts byadding a vertical ribbon strip to , with coefficient
Problem. Find a combinatorial formula forfor general and .
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General LLT Polynomials
Two things to prove...Second,
Theorem (Grojnowski, H.). The matrix coefficient
(the coefficient of in ) isalways positive, i.e., in .
May 28, 2007 62
General LLT Polynomials
Theorem (Grojnowski, H.). The matrix coefficient
(the coefficient of in ) isalways positive, i.e., in .
which is a corollary to—
Theorem (G & H). Given any (possibly infinite) Weylgroup and parabolic , define ,where , and . The matrixcoefficients of a KL basis element , acting by leftmultiplication on the basis of , arepositive.
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General LLT Polynomials
Theorem (G & H). Given any (possibly infinite) Weylgroup and parabolic , define ,where , and . The matrixcoefficients of a KL basis element , acting by leftmultiplication on the basis of , arepositive.
Proof uses the standard picture of as convolutionalgebra of MHM´s or étale perverse sheaves (take yourpick) on the flag variety...
...plus a maximally delicate variant of the usualreasoning [Springer, Lusztig], using purity of hyperbolicrestriction [Braden, others].
May 28, 2007 64
General LLT Polynomials
The Moral
Although many KLpolynomials are mean...
May 28, 2007 65
General LLT Polynomials
The Moral
Although many KLpolynomials are mean...
...some are friendly!