Math 2.0

Combinatorics Definition

Macdonald polynomial

Builds on Symmetric functionPower sum symmetric polynomialInteger partition

Definition

Let Λ\Lambda be the ring of symmetric functions in variables x=(x1,x2,… )x = (x_1, x_2, \dots) with coefficients in the field ℚ(q,t)\Q(q, t). For a partition λ\lambda, let mλm_\lambda be the monomial symmetric function and let pλ=pλ1pλ2⋯p_\lambda = p_{\lambda_1} p_{\lambda_2} \cdots be the product of power sums. Let zλ=∏i≥1imimi!z_\lambda = \prod_{i \ge 1} i^{m_i} m_i!, where mim_i is the number of parts of λ\lambda equal to ii. Define a bilinear form on Λ\Lambda by ⟨pλ,pμ⟩q,t=δλμ zλ∏i1−qλi1−tλi.\ip{p_\lambda}{p_\mu}_{q,t} = \delta_{\lambda\mu} \, z_\lambda \prod_{i} \frac{1 - q^{\lambda_i}}{1 - t^{\lambda_i}}. For partitions λ,μ\lambda, \mu of the same number, write μ≤λ\mu \le \lambda (dominance order) if μ1+⋯+μk≤λ1+⋯+λk\mu_1 + \dots + \mu_k \le \lambda_1 + \dots + \lambda_k for all kk. The Macdonald polynomials are the unique family (Pλ(x;q,t))λ(P_\lambda(x; q, t))_\lambda in Λ\Lambda such that:

  1. Pλ=mλ+∑μ<λcλμmμP_\lambda = m_\lambda + \sum_{\mu < \lambda} c_{\lambda\mu} m_\mu for some cλμ∈ℚ(q,t)c_{\lambda\mu} \in \Q(q, t).
  2. ⟨Pλ,Pμ⟩q,t=0\ip{P_\lambda}{P_\mu}_{q,t} = 0 for λ≠μ\lambda \ne \mu.

Remarks

The existence and uniqueness of this family is a theorem of Macdonald; it is not automatic, because dominance is only a partial order.

Example: for λ=(2)\lambda = (2), P(2)=m(2)+(1+q)(1−t)1−qtm(1,1)P_{(2)} = m_{(2)} + \frac{(1+q)(1-t)}{1-qt} m_{(1,1)}. Check: m(2)=p2m_{(2)} = p_2 and m(1,1)=12(p12−p2)m_{(1,1)} = \tfrac12 (p_1^2 - p_2), and z(2)=z(1,1)=2z_{(2)} = z_{(1,1)} = 2. Then ⟨p2,m(1,1)⟩q,t=−1−q21−t2\ip{p_2}{m_{(1,1)}}_{q,t} = -\frac{1-q^2}{1-t^2} and ⟨m(1,1),m(1,1)⟩q,t=(1−q)(1−qt)(1−t)(1−t2)\ip{m_{(1,1)}}{m_{(1,1)}}_{q,t} = \frac{(1-q)(1-qt)}{(1-t)(1-t^2)}, so ⟨P(2),P(1,1)⟩q,t=0\ip{P_{(2)}}{P_{(1,1)}}_{q,t} = 0 with P(1,1)=m(1,1)P_{(1,1)} = m_{(1,1)}.

Non-example: the Schur function s(2)=m(2)+m(1,1)s_{(2)} = m_{(2)} + m_{(1,1)} satisfies clause 1 but not clause 2 when q≠tq \ne t. Indeed ⟨s(2),m(1,1)⟩q,t=(1−q)(t−q)(1−t)(1−t2)≠0\ip{s_{(2)}}{m_{(1,1)}}_{q,t} = \frac{(1-q)(t-q)}{(1-t)(1-t^2)} \ne 0.

At q=tq = t the form is the Hall inner product and Pλ=sλP_\lambda = s_\lambda. At q=0q = 0, PλP_\lambda is the Hall–Littlewood polynomial Pλ(x;t)P_\lambda(x; t).

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Integer partition, concept, no evidence levelInteger partitionconceptPower sum symmetric polynomial, concept, no evidence levelPower sum symmetric p…conceptSymmetric function, concept, no evidence levelSymmetric functionconceptMacdonald polynomial, definition, no evidence levelMacdonald polynomialdefinition
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  1. Integer partition concept
  2. Power sum symmetric polynomial concept
  3. Symmetric function concept
  4. Macdonald polynomial definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Macdonald_polynomials
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Kinddefinition
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MSC05E05, 33D52
DefinesMacdonald polynomial
UsesSymmetric function, Power sum symmetric polynomial, Integer partition
Changed2 November 2026

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