Macdonald Identity Research Articles

Theta Functions and Brownian Motion

A theta function for an arbitrary connected and simply connected compact simple Lie group is defined as an infinite determinant that is naturally related to the transformation of a family of independent Gaussian random variables associated with a pinned Brownian motion in the Lie group. From this definition of a theta function, the equality of the product and the sum expressions for a theta function is obtained. This equality for an arbitrary connected and simply connected compact simple Lie group is known as a Macdonald identity which generalizes the Jacobi triple product for the elliptic theta function associated with su(2).

An extension of Macdonald’s identity for $${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$$

Let $n$ be an odd positive integer. In this short elementary note, we slightly extend Macdonald's identity for $\mathfrak{sl}_{n}$ into a two-variables identity in the spirit of Jacobi forms. The peculiarity of this work lies in its proof which uses Wronskians of vector-valued $\theta$-functions. This complements the work of A. Milas towards modular Wronskians and denominator identities.

Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism

We study the derived representation scheme DRep _n(A) parametrizing the n -dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep _n(A) is isomorphic to the Chevalley–Eilenberg homology of the current Lie coalgebra \mathfrak {gl}_n^*(\bar{C}) defined over a Koszul dual coalgebra of A . This gives a conceptual explanation to some of the main results of [BKR] and [BR], relating them (via Koszul duality) to classical theorems on (co)homology of current Lie algebras \mathfrak {gl}_n(A) . We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra \mathfrak g , we define the derived affine scheme DRep _{\mathfrak g}(\mathfrak a) parametrizing the representations (in \mathfrak g ) of a Lie algebra \mathfrak{a} ; we show that the homology of DRep_ {\mathfrak g}(\mathfrak a) is isomorphic to the Chevalley–Eilenberg homology of the Lie coalgebra \mathfrak g^*(\bar{C}) , where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra \mathfrak a . We construct a canonical DG algebra map \Phi_{\mathfrak g}(\mathfrak a): \mathrm {DRep}_{\mathfrak g}(\mathfrak a)^G \to \mathrm {DRep}_{\mathfrak h}(\mathfrak a)^W , relating the G -invariant part of representation homology of a Lie algebra \mathfrak a in \mathfrak g to the W -invariant part of representation homology of \mathfrak a in a Cartan subalgebra of \mathfrak g . We call this map the derived Harish-Chandra homomorphism as it is a natural homological extension of the classical Harish-Chandra restriction map. We conjecture that, for a two-dimensional abelian Lie algebra \mathfrak{a} , the derived Harish-Chandra homomorphism is a quasi-isomorphism. We provide some evidence for this conjecture, including proofs for \mathfrak {gl}_2 and \mathfrak {sl}_2 as well as for \mathfrak {gl}_n, \mathfrak {sl}_n, \mathfrak{so}_n and \mathfrak{sp}_{2n} in the inductive limit as n \to \infty . For any complex reductive Lie algebra \mathfrak g , we compute the Euler characteristic of DRep _{\mathfrak g}(\mathfrak a)^G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep _{\mathfrak h}(\mathfrak a)^W . This yields an interesting combinatorial identity, which we prove for \mathfrak {gl}_n and \mathfrak{sl}_n (for all n ). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proposed in [Ha1, F] and proved in [FGT]. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.

A Nekrasov–Okounkov type formula for [formula omitted

In 2008, Han rediscovered an expansion of powers of Dedekind η function attributed to Nekrasov and Okounkov (which was actually first proved the same year by Westbury) by using a famous identity of Macdonald in the framework of type A˜ affine root systems. In this paper, we obtain new combinatorial expansions of powers of η, in terms of partition hook lengths, by using the Macdonald identity in type C˜ and a new bijection between vectors with integral coordinates and a subset of t-cores for integer partitions. As applications, we derive a symplectic hook formula and an unexpected relation between the Macdonald identities in types C˜, B˜, and BC˜. We also generalize these expansions through the Littlewood decomposition and deduce in particular many new weighted generating functions for subsets of integer partitions and refinements of hook formulas.

A Nekrasov-Okounkov type formula for affine $\widetilde{C}$

In 2008, Han rediscovered an expansion of powers of Dedekind $\eta$ function due to Nekrasov and Okounkov by using Macdonald's identity in type $\widetilde{A}$. In this paper, we obtain new combinatorial expansions of powers of $\eta$, in terms of partition hook lengths, by using Macdonald's identity in type $\widetilde{C}$ and a new bijection. As applications, we derive a symplectic hook formula and a relation between Macdonald's identities in types $\widetilde{C}$, $\widetilde{B}$, and $\widetilde{BC}$. En 2008, Han a redécouvert un développement des puissances de la fonction $\eta$ de Dedekind, dû à Nekrasov et Okounkov, en utilisant l’identité de Macdonald en type $\widetilde{A}$. Dans cet article, nous obtenons un nouveau développement combinatoire des puissances de $\eta$, en termes de longueurs d’équerres de partitions, en utilisant l’identité de Macdonald en type $\widetilde{C}$ ainsi qu’une nouvelle bijection. Plusieurs applications en sont déduites, comme un analogue symplectique de la formule des équerres, ou une relation entre les identités de Macdonald en types $\widetilde{C}$, $\widetilde{B}$, et $\widetilde{BC}$.

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobiʼs triple product identity and the t-coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewellʼs and Chen–Chen–Huangʼs octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system A2 as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.

Macdonald?s Identities and the Large N Limit of YM 2 on the Cylinder

We give a rigorous calculation of the large N limit of the partition function of SU(N) gauge theory on a 2D cylinder in the case where one boundary holomony is a so-called special element of type $\rho$. By MacDonald's identity, the partition function factors in this case as a product over positive roots and it is straightforward to calculate the large N asymptotics of the free energy. We obtain the unexpected result that the free energy in these cases is asymptotic to N times a functional of the limit densities of eigenvalues of the boundary holonomies. This appears to contradict the predictions of Gross-Matysin and Kazakov-Wynter that the free energy should have a limit governed by the complex Burgers equation.

The homology of graded infinite-dimensional Lie algebras in connection with Macdonald identities

D. Fucks' monograph devoted to the cohomology of infinite-dimensional Lie algebras contains an error in calculating the homology of a graded affine Kac-Moody algebra of type A n (1) , so that the proof of the corresponding Macdonald identity, which is based on that calculation, is incorrect. In the present paper, a revised proof is suggested.

A new proof of the Macdonald identities for An−1

AbstractA new, elementary proof of the Macdonald identities for An−1 using induction on n is given. Specifically, the Macdonald identity for An is deduced by multiplying the Macdonald identity for An−1 and n Jacobi triple product identities together.

The Schur function realization of vertex operators

We present a realization of untwisted vertex operators in terms of operations on Schur functions. Calculations of matrix elements and traces of products of vertex operators are performed using results from the classical theory of symmetric functions. The concepts of compound, composite and supersymmetric Schur functions naturally appear in this context. Furthermore, a trace formula for a product of vertex operators turns out to be a generalization of a MacDonald identity reformulated in terms of Schur functions.

Vertex operator algebras associated to representations of affine and Virasoro algebras

The first construction of the integrable highest-weight representations of affine Lie algebras or loop algebras by Kac i-K] was greatly inspired by the generalization of the Weyl denominator formula for affine roots systems discovered earlier by Macdonald [M]. Though the Macdonald identity found its natural context in representation theory, its mysterious modular invariance was not understood until the work of Witten [W-I on the geometric realization of representations of the loop groups corresponding to loop algebras. The work of Witten clearly indicated that the representations of loop groups possess a very rich structure of conformal field theory which appeared in physics literature in the work of Belavin, Polyakov, and Zamolodchikov [BPZ-I. Independently (though two years later), Borcherds, in an attempt to find a conceptual understanding of a certain algebra of vertex operators invariant under the Monster [FLM1], introduced in [B-I a new algebraic structure. We call vertex operator algebras a slightly modified version of Borcherd’s new algebras [FLM2].

A Summation Theorem for Hypergeometric Series Very-Well-Poised on $G_2 $

An analogue of Bailey’s ${}_6 \psi _6 $ summation theorem is proved for basic hypergeometric series that are very well poised on the Lie algebra $G_2 $. As a limiting case, a new proof of the Macdonald identity associated to the affine root system of type $G_2 $ is obtained. A summation theorem for ordinary hypergeometric series that are very well poised on $G_2 $ is proved by Carlson’s theorem.

A generalization of the Kostant—Macdonald identity

Let X be a smooth projective variety admitting an algebraic vector field V with exactly one zero and a holomorphic C(*)-action lambda so that the condition dlambda(t).V = t(p)V holds for all t in C(*). The purpose of this note is to report on a product formula for the Poincaré polynomial of X which specializes to the classical identity [Formula: see text] when X is the flag variety of a semisimple complex Lie group. A surprising corollary is that the second Betti number of such an X is the multiplicity of largest weight of the linear C(*)-action on the tangent space of X at the sink of lambda. We discuss several examples, including a construction of the rational Fano 3-folds A'(22) and B(5) which is due to Konarski [Konarski, J. (1989) in C.M.S. Conference Proceedings, ed. Russell, P. (Am. Math. Soc., Providence, RI), in press].