[Paper Review] Hochschild homology of certain Soergel bimodules

Marko Stošić|ArXiv.org|Oct 20, 2008

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper computes the Hochschild homology of specific Soergel bimodules, particularly $ R_{k,l} igotimes_{k+l} R_{k,l} $, and establishes the existence and uniqueness of a graded bimodule map of degree $ 2kl $, which is essential for categorifying the colored HOMFLY-PT polynomial. The key result is a complete computation of the Hochschild homology, providing a categorification of the digon move axiom in the calculus of colored HOMFLY-PT invariants via Koszul complexes and Schur polynomials.

ABSTRACT

In this paper we compute Hochschild homology of certain Soergel bimodules. Moreover, we describe explicitly the graded bimodule maps between Soergel bimodules. This computations are motivated by the categorifications of the colored HOMFLY-PT polynomial for links via Hochschild homology of Soergel bimodules.

Motivation & Objective

• To compute the Hochschild homology of the Soergel bimodule $ R_{k,l} igotimes_{k+l} R_{k,l} $, which arises in the categorification of the colored HOMFLY-PT polynomial.
• To establish the existence and uniqueness of a graded bimodule map from $ R_{k,l} $ to $ R_{k,l} igotimes_{k+l} R_{k,l} $ of degree $ 2kl $, and prove no such maps exist in lower degrees.
• To provide an algebraic foundation for the invariance of Khovanov-Rozansky link invariants under Reidemeister moves, particularly in the 2-colored case.
• To relate the structure of the bimodule map to orthogonality in Frobenius algebras appearing in Khovanov-Lauda categorified quantum groups.

Proposed method

• Uses the Koszul complex model to reduce Hochschild homology computation to the homology of polynomial rings invariant under symmetric group actions.
• Represents Soergel bimodules via quotient rings of polynomial rings, with generators given by elementary and Schur polynomials.
• Applies Giambelli’s determinantal formula to express Schur polynomials in terms of elementary symmetric polynomials.
• Employs a formal expression involving quantum integers and $ q $-dimensions to compute the graded dimensions of homology groups.
• Constructs a free graded resolution of ideals in the quotient ring to compute $ \mathrm{qdim}(I_j) $, essential for the final identity.
• Uses combinatorial identities on $ q $-binomial coefficients and Kronecker delta simplifications to verify the categorified digon move axiom.

Experimental results

Research questions

• RQ1 Is there a unique graded bimodule map of degree $ 2kl $ from $ R_{k,l} $ to $ R_{k,l} \bigotimes_{k+l} R_{k,l} $, and are there no such maps of lower degree?
• RQ2 How does the Hochschild homology of $ R_{k,l} \bigotimes_{k+l} R_{k,l} $ decompose, and what is its $ q $-dimensional structure?
• RQ3 Can the computation of Hochschild homology be used to categorify the digon move axiom in the colored HOMFLY-PT polynomial calculus?
• RQ4 What is the role of the unique bimodule map in establishing orthogonality in the Frobenius algebra used in Khovanov-Lauda categorified quantum groups?
• RQ5 Does the Koszul complex model yield a complete and computable description of the homology for these bimodules?

Key findings

• The bimodule map from $ R_{k,l} $ to $ R_{k,l} \bigotimes_{k+l} R_{k,l} $ of degree $ 2kl $ exists and is unique, with no nonzero maps of lower degree.
• The Hochschild homology of $ R_{k,l} \bigotimes_{k+l} R_{k,l} $ is computed explicitly, with homology groups $ H_{-i} $ isomorphic to direct sums of ideals in the quotient ring $ R' \simeq R^\prime $.
• The $ q $-dimension of the homology satisfies the identity $ \sum_{i=0}^{l} (-1)^i t^i \mathrm{qdim} H_{-i} = \mathrm{qdim} R' \cdot \prod_{i=1}^{l} (1 - t^{-1} q^{2k + 2i - 1}) $, confirming the digon move axiom.
• The computation of $ \mathrm{qdim}(I_j) $ is achieved via a free graded resolution, yielding $ \mathrm{qdim} I_j = \mathrm{qdim} R' \sum_{i=j}^{l} (-1)^{i-j} q^{i(i+1+2(k-l)) + j(j-1)} \binom{i-1}{i-j}_q \binom{l}{i}_q $.
• The final identity is verified through combinatorial simplifications involving $ q $-binomial coefficients and Kronecker delta identities, confirming the categorified relation.
• The unique bimodule map plays a structural role in the orthogonality of a basis in the Frobenius algebra used in Khovanov-Lauda's categorified quantum group calculus.