Idempotents of Hecke Algebras of Type A

Abstract. The colored HOMFLY polynomial is the quantum invariant of oriented links in S 3 associated with irreducible representations of the quantum group U q (sl N ). In this paper, using an approach to calculate quantum invariants of links via the cabling-projection rule, we derive a formula for the colored HOMFLY polynomial in terms of the characters of the Hecke algebras and Schur polynomials. The technique leads to a fairly simple formula for the colored HOMFLY polynomial of torus links. This formula allows us to test the Labastida-Mariño-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

show abstract

“…See, for instance, [1], [4], [6], [15]. For the reader's convenience, we present a possible choice of them as follows (see [1,Theorem 4.7]). …”

Section: Hecke Algebras and Colored Homfly Polynomialmentioning

confidence: 99%

On the Hecke algebras and the colored HOMFLY polynomial

Lin

Zheng²

2009

Trans. Amer. Math. Soc.

101

View full text Add to dashboard Cite

show abstract

“…The left hand sides of the previous two equations are equal because a n is central. Hence, α n ρ(a n ) = ρ(α n )a n and thus ρ [2] constructed idempotents E λ in R n n for any Young diagram λ with n cells. To do this, they considered a three-dimensional version of R n n where the distinguished boundary points are lined up along the cells of the Young diagram λ.…”

Section: Lemma 32 ρmentioning

confidence: 99%

“…Blanchet describes in [3] an explicit semi-simple decomposition of R n n . The lemmas he uses for his construction are also contained in [2] as explained in…”

Section: Theorem 72mentioning

confidence: 99%

See 1 more Smart Citation

Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

Lukac¹

2005

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

The Hecke algebra H n contains well known idempotents E λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja [5].A skein theoretical description of E λ was given by Aiston and Morton [2]. The closure of E λ becomes an element Q λ of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s λ as stated in theorem 8.2. But previous proofs of this fact as in [1] used results about quantum groups which were far beyond the scope of skein theory.Our elementary proof was motivated by [6] and uses only skein theory and basic algebra. The skein of a planar surfaceWe consider a planar surface F with designated n incoming and n outgoing boundary points for some integer n ≥ 0. The Homfly skein S(F ) is defined as the module of linear combinations of oriented tangles in F quotiented by regular isotopy (i.e. Reidemeister moves II and III), the two local skein relations in figure 1 where v and s are variables, and the relation that a disjoint simple closed curve can be removed from a diagram at the expense of multiplication with the scalar δ =

show abstract

“…If c has index (i, j) (i-th row, and j-th column), then the corresponding point in D 2 is j+i √ −1 n+1 . Following Aiston and Morton [1], we can define in H ✷ λ a minimal idempotent y λ which is a version of the corresponding Young idempotent of the symmetric group algebra. The idea of the construction is to insert symmetrizers along rows and antisymmetrizers along columns, and then to normalize.…”

Section: Hecke Algebrasmentioning

confidence: 99%

Skein construction of idempotents in Birman-Murakami-Wenzl algebras

Beliakova¹,

Blanchet²

2001

Mathematische Annalen

View full text Add to dashboard Cite

Abstract. We give skein theoretic formulas for minimal idempotents in the Birman-Murakami-Wenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula for quantum dimensions is given. This proof does not use the representation theory of quantum groups and the character formulas.

show abstract

Idempotents of Hecke Algebras of Type A

Cited by 62 publications

References 13 publications

On the Hecke algebras and the colored HOMFLY polynomial

On the Hecke algebras and the colored HOMFLY polynomial

Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

Skein construction of idempotents in Birman-Murakami-Wenzl algebras

Contact Info

Product

Resources

About