Concha–Lapointe Identity in Macdonald Theory

• Concha–Lapointe Identity is a duality theorem for partially symmetric Macdonald polynomials, featuring parameter inversion, variable reversal, and a q-shift in the nonsymmetric block.
• It employs combinatorial proofs via non-attacking augmented fillings and utilizes Demazure–Lusztig operators to manage transformations between symmetric and nonsymmetric bases.
• The refined identity extends to permuted basement Macdonald polynomials and connects with the Kazhdan–Lusztig involution, deepening our understanding of duality in algebraic combinatorics.

The Concha–Lapointe identity is an identity for partially symmetric Macdonald polynomials that extends the classical invariance

$P_\lambda(x;q^{-1},t^{-1})=P_\lambda(x;q,t)$

of ordinary symmetric Macdonald polynomials. In the form studied in "Combinatorial proof of a permuted basement Macdonald polynomial identity" (Orr et al., 28 Aug 2025), it governs the effect of simultaneously inverting $(q,t)$, reversing the nonsymmetric variables, applying a $q$-shift to those variables, and acting by a Demazure–Lusztig operator. The same work refines the identity to a subfamily of Alexandersson’s permuted basement Macdonald polynomials and proves that the identity is equivalent to the assertion that normalized partially symmetric Macdonald polynomials are fixed under the Kazhdan–Lusztig involution (Orr et al., 28 Aug 2025).

1. Defining formula

In the notation of (Orr et al., 28 Aug 2025), let $0\le m\le n$, let $\lambda\in(\mathbb Z_{\ge 0})^m$ be a partition with

$\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$

and let $$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$ be a composition. If $P_{\lambda\mid\gamma}(x;q,t)$ denotes the partially symmetric Macdonald polynomial, $\omega_0^{[m+1,n]}$ is the long element in the symmetric group on $\{m+1,\dots,n\}$, $(q,t)$0 is the corresponding Demazure–Lusztig operator, and

$(q,t)$1

then the Concha–Lapointe identity is

$(q,t)$2

Here $(q,t)$3 (Orr et al., 28 Aug 2025).

Two structural features distinguish this formula from the ordinary symmetric case. First, parameter inversion is no longer a literal fixed-point statement. Second, the nonsymmetric block $(q,t)$4 is both reversed and multiplied by $(q,t)$5. The paper emphasizes that this shift is a genuine feature of the partially symmetric setting: when $(q,t)$6 it disappears, while when $(q,t)$7 homogeneity removes the global $(q,t)$8-factor and one recovers the usual nonsymmetric duality formula (Orr et al., 28 Aug 2025).

2. Algebraic setting and partially symmetric Macdonald polynomials

The identity sits between the symmetric and nonsymmetric Macdonald theories. For a partition $(q,t)$9 with $q$0, the ordinary Macdonald polynomial is

$q$1

For a composition $q$2, the nonsymmetric Macdonald polynomial is

$q$3

The partially symmetric theory fixes $q$4 and imposes symmetry only in the first $q$5 variables (Orr et al., 28 Aug 2025).

Let $q$6. The partially symmetric Macdonald polynomial is defined by

$q$7

where $q$8 denotes the minimal coset representatives for the stabilizer of $q$9 inside $0\le m\le n$0 (Orr et al., 28 Aug 2025). This exhibits $0\le m\le n$1 as an interpolation object: it is symmetric in $0\le m\le n$2, but no symmetry is imposed on $0\le m\le n$3.

The operators $0\le m\le n$4 are built from the Demazure–Lusztig generators. For $0\le m\le n$5, if $0\le m\le n$6, then

$0\le m\le n$7

For a reduced expression $0\le m\le n$8, one sets

$0\le m\le n$9

and this is independent of the reduced expression. A basic compatibility used throughout the theory is that if $\lambda\in(\mathbb Z_{\ge 0})^m$0, then

$\lambda\in(\mathbb Z_{\ge 0})^m$1

(Orr et al., 28 Aug 2025).

A plausible implication is that the Concha–Lapointe identity should be viewed less as an isolated transformation formula than as a structural duality theorem for the parabolic interpolation between symmetric and nonsymmetric Macdonald bases.

3. Refinement to permuted basement Macdonald polynomials

The 2025 paper does not stop at partially symmetric polynomials. It refines the Concha–Lapointe identity to a subfamily of Alexandersson’s permuted basement Macdonald polynomials (Orr et al., 28 Aug 2025).

For $\lambda\in(\mathbb Z_{\ge 0})^m$2 and $\lambda\in(\mathbb Z_{\ge 0})^m$3, define

$\lambda\in(\mathbb Z_{\ge 0})^m$4

where

$\lambda\in(\mathbb Z_{\ge 0})^m$5

The paper compares this normalization with Alexandersson’s notation $\lambda\in(\mathbb Z_{\ge 0})^m$6 through

$\lambda\in(\mathbb Z_{\ge 0})^m$7

The relevant family is the one for which the basement permutation is a concatenation

$\lambda\in(\mathbb Z_{\ge 0})^m$8

with $\lambda\in(\mathbb Z_{\ge 0})^m$9 a permutation of $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$0 and $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$1 a permutation of $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$2, so there is no mixing between the two blocks (Orr et al., 28 Aug 2025).

If $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$3 are the complements within their respective consecutive alphabets, and

$\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$4

then the refined identity is

$\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$5

The paper proves that the original Concha–Lapointe identity is recovered by specializing to $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$6, taking $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$7 on $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$8, and summing over $\lambda_1\ge \lambda_2\ge \cdots \ge \lambda_m,$9 (Orr et al., 28 Aug 2025).

This refinement makes clear that the identity is not merely a statement about a single basis element $$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$0. It is a blockwise complement symmetry in a larger permuted-basement framework.

4. Combinatorial proof by non-attacking augmented fillings

The combinatorial proof is based on the Haglund–Haiman–Loehr filling formula, extended by Alexandersson to the permuted basement setting (Orr et al., 28 Aug 2025). For a composition $$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$1, the column diagram is

$$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$2

and the augmented diagram is

$$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$3

A basement permutation $$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$4 is inserted by setting

$$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$5

The relevant fillings are the non-attacking augmented fillings $$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$6 (Orr et al., 28 Aug 2025).

The proof is organized around a bijection

$$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$7

that replaces entries in $$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$8 according to $$\gamma=(\gamma_1,\dots,\gamma_{n-m})\in(\mathbb Z_{\ge 0})^{n-m}$$9 and entries in $P_{\lambda\mid\gamma}(x;q,t)$0 according to $P_{\lambda\mid\gamma}(x;q,t)$1. The decisive point is not only that this is a bijection, but that it transforms the HHL statistics in exactly the way needed for the $P_{\lambda\mid\gamma}(x;q,t)$2- and $P_{\lambda\mid\gamma}(x;q,t)$3-weights (Orr et al., 28 Aug 2025).

Two identities drive the argument. If

$P_{\lambda\mid\gamma}(x;q,t)$4

then

$P_{\lambda\mid\gamma}(x;q,t)$5

and

$P_{\lambda\mid\gamma}(x;q,t)$6

The first relation explains the appearance of both the global factor $P_{\lambda\mid\gamma}(x;q,t)$7 and the $P_{\lambda\mid\gamma}(x;q,t)$8-shift on the second block of variables. The second shows that the complement operation preserves the $P_{\lambda\mid\gamma}(x;q,t)$9-weight after parameter inversion (Orr et al., 28 Aug 2025).

The paper also reformulates the proof in terms of inversion triples and coinversion triples. In that language, $\omega_0^{[m+1,n]}$0 equals the number of inversion triples and $\omega_0^{[m+1,n]}$1 equals the number of coinversion triples, while coinversion triples in non-attacking fillings are characterized by cyclic ordering of entries. This triple calculus is the local mechanism by which the complement map converts the $\omega_0^{[m+1,n]}$2 statistics into the $\omega_0^{[m+1,n]}$3 statistics (Orr et al., 28 Aug 2025).

5. Kazhdan–Lusztig involution and normalization

One of the central conceptual results is that the Concha–Lapointe identity is equivalent to Kazhdan–Lusztig self-duality for normalized partially symmetric Macdonald polynomials (Orr et al., 28 Aug 2025).

Let

$\omega_0^{[m+1,n]}$4

The Kazhdan–Lusztig involution on the positive affine Hecke algebra induces a $\omega_0^{[m+1,n]}$5-linear involution on $\omega_0^{[m+1,n]}$6: $\omega_0^{[m+1,n]}$7 where $\omega_0^{[m+1,n]}$8 is the long element of $\omega_0^{[m+1,n]}$9 (Orr et al., 28 Aug 2025).

For nonsymmetric Macdonald polynomials, the normalized object

$\{m+1,\dots,n\}$0

satisfies

$\{m+1,\dots,n\}$1

For partially symmetric polynomials, the corresponding theorem is

$\{m+1,\dots,n\}$2

where

$\{m+1,\dots,n\}$3

Hence the normalized partially symmetric polynomial

$\{m+1,\dots,n\}$4

satisfies

$\{m+1,\dots,n\}$5

The paper proves that this fixed-point statement is equivalent to the Concha–Lapointe identity (Orr et al., 28 Aug 2025).

This equivalence places the identity in the standard Hecke-theoretic architecture of Macdonald theory. A plausible implication is that the identity is best interpreted as a duality theorem in a maximal parabolic module, rather than only as a transformation rule for variables and parameters.

6. Specializations, examples, applications, and terminological scope

The two extremal specializations recover the familiar endpoints of Macdonald theory. When $\{m+1,\dots,n\}$6, the composition $\{m+1,\dots,n\}$7 is empty, there is no $\{m+1,\dots,n\}$8-shifted block, and the Concha–Lapointe identity reduces to

$\{m+1,\dots,n\}$9

When $(q,t)$00, $(q,t)$01, and the identity becomes the known nonsymmetric duality after using homogeneity to remove the global $(q,t)$02-factor (Orr et al., 28 Aug 2025).

The paper also records an explicit example in the refined setting. For $(q,t)$03, $(q,t)$04, basement permutations

$(q,t)$05

their complements are

$(q,t)$06

With $(q,t)$07, one has

$(q,t)$08

and the refined identity becomes

$(q,t)$09

(Orr et al., 28 Aug 2025).

The identity is not presented as an isolated combinatorial curiosity. The abstract states that it “played a key role” in work of Bechtloff Weising and Orr linking partially symmetric Macdonald polynomials to parabolic flag Hilbert schemes, and the paper situates it in the broader program of extending Macdonald positivity to partially symmetric settings; the analogous positivity statement is described there as remaining open (Orr et al., 28 Aug 2025).

In the supplied arXiv sources, the name Concha–Lapointe identity appears explicitly in the partially symmetric Macdonald-polynomial setting just described. Other papers in the set treat unrelated identities and explicitly do not use the term: the hydrogen identity for Laplacians (Knill, 2018), the identity of Chaundy and Bullard (Aharonov et al., 2013), a family of orthogeodesic identities extending Basmajian’s identity (Basmajian et al., 2020), and an unnamed identity involving symmetric polynomials and Lagrangian Grassmannians (Hiep, 2016). This suggests that, within the present corpus, the established mathematical referent of the term is the Macdonald-theoretic duality formula and its permuted-basement refinement.