Mixed Macdonald Dimensions

• Mixed Macdonald dimensions are refined invariants defined as dual evaluations of Macdonald polynomials over an admissible pair of root systems, capturing distinct q and t gradings.
• They generalize quantum dimensions by factorizing explicit products and incorporating dual refined Weyl data, thereby extending representation-theoretic interpretations to non-simply laced sectors.
• These dimensions play a pivotal role in frameworks like Vogel universality, superspace evaluations, and probabilistic Macdonald processes, linking combinatorial, algebraic, and geometric insights.

Searching arXiv for recent and foundational papers on mixed Macdonald dimensions and related Macdonald-dimension frameworks. Mixed Macdonald dimensions arise most specifically as factorized values of root-system Macdonald polynomials attached to an admissible pair $(R,S)$ of different root systems, evaluated at the dual refined Weyl point $x=q^{2r_k^*}$ (Bishler, 15 Jul 2025). In a broader but closely related usage, the same expression names a family of $q,t$-refined dimension-like invariants in which two sectors are coupled: two root systems, two partitions, bosonic and fermionic data, bulk and boundary weights, or partition and charge variables (Bishler et al., 22 May 2025). Across these settings, the common feature is a factorized or explicitly computable quantity that generalizes quantum dimensions by replacing Schur characters with Macdonald objects and by retaining separate $q$- and $t$-gradings.

1. Root-system definition

For an admissible pair of root systems $(R,S)$, Macdonald polynomials

$P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$

are indexed by dominant weights $\lambda$ of $R$, characterized by triangularity with respect to $m_\lambda^R$ and orthogonality with respect to the Macdonald scalar product built from the Macdonald density $x=q^{2r_k^*}$0 (Bishler, 15 Jul 2025). The refined Weyl data are

$x=q^{2r_k^*}$1

with $x=q^{2r_k^*}$2 and $x=q^{2r_k^*}$3.

The paper distinguishes two special evaluations. The first is the Macdonald dimension

$x=q^{2r_k^*}$4

which refines the ordinary quantum dimension. The second is the dual Macdonald dimension

$x=q^{2r_k^*}$5

for which Macdonald’s evaluation theorem yields a product formula (Bishler, 15 Jul 2025): $x=q^{2r_k^*}$6

In the terminology of that paper, mixed Macdonald dimensions are precisely these dual Macdonald dimensions for $x=q^{2r_k^*}$7. The defining feature is that the polynomial is attached to $x=q^{2r_k^*}$8, while the deformation parameters $x=q^{2r_k^*}$9 and the evaluation point $q,t$0 incorporate the second root system $q,t$1 through the factors $q,t$2 such that $q,t$3 (Bishler, 15 Jul 2025).

A structural distinction is that $q,t$4 does not in general factorize, whereas $q,t$5 does. For simply laced $q,t$6, one has $q,t$7, hence $q,t$8; this coincidence is what makes Vogel-type universality possible in the simply laced case (Bishler, 15 Jul 2025).

2. Mixed pairs and explicit factorization

The prototypical mixed pairs analyzed explicitly are

$q,t$9

all sharing the Weyl group of type $q$0 but differing in the assignment of $q$1, $q$2, and $q$3 (Bishler, 15 Jul 2025). In each case the dual refined Weyl vector

$q$4

mixes data from both root systems, and the resulting factorized expression depends simultaneously on both systems.

For $q$5, long roots satisfy $q$6, $q$7, $q$8, while short roots satisfy $q$9, $t$0, $t$1. For $t$2, the long-root sector instead uses $t$3, $t$4, $t$5. The non-reduced $t$6 cases introduce separate parameters $t$7, $t$8, and $t$9, and the factorization point becomes $(R,S)$0 in both $(R,S)$1 and $(R,S)$2 (Bishler, 15 Jul 2025).

The explicit computations in these four series are carried out for the adjoint representation or the representation $(R,S)$3, depending on the pair. The resulting formulas are finite products of $(R,S)$4 factors and admit controlled limits as $(R,S)$5 and $(R,S)$6. The unrefined limit $(R,S)$7 recovers products and ratios of $(R,S)$8-numbers matching quantum dimensions of $(R,S)$9, $P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$0, or combinations thereof, while the classical limit $P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$1 returns ordinary dimensions (Bishler, 15 Jul 2025).

Conceptually, these mixed dimensions occupy the non-simply-laced sector in which the factorization point is intrinsically dual. This suggests that the mixed theory is not a peripheral variant but the natural habitat of factorized Macdonald evaluations once one leaves the simply laced regime.

3. Universality, Macdonald Littlewood–Richardson coefficients, and the adjoint sector

A second, distinct use of the phrase appears in the refined Vogel-universality program. There the central claim is that bare Macdonald dimensions are generally not Vogel-universal objects, whereas certain mixed combinations are: products of Macdonald dimensions with Macdonald Littlewood–Richardson coefficients, and further sums organized by universally-irreducible representations, or uirreps (Bishler et al., 22 May 2025).

For the $P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$2-series one writes

$P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$3

with $P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$4 the Macdonald Littlewood–Richardson coefficients. The mixed quantities are then

$P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$5

The paper argues that these, rather than $P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$6 alone, are the objects that admit Vogel-type universal formulas in the simply laced sector (Bishler et al., 22 May 2025).

The basic case is the adjoint square. The universal decomposition is

$P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$7

and the corresponding mixed quantities are denoted

$P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$8

with analogous terms for $P^{(R,S)}_{\lambda}(x\,|\,t_\alpha\,|\,q,t)$9, $\lambda$0, $\lambda$1, and $\lambda$2 (Bishler et al., 22 May 2025). These are then expressed as universal rational functions of

$\lambda$3

This universal mixed sector is motivated by refined Chern–Simons theory and hyperpolynomials. The Hopf-link and torus-link $\lambda$4 formulas are sums of universal framing factors multiplied by the mixed objects $\lambda$5, $\lambda$6, $\lambda$7, $\lambda$8, $\lambda$9, and $R$0 (Bishler et al., 22 May 2025). In this framework, mixed Macdonald dimensions are the refined observables that survive passage from ordinary representation theory to the universal adjoint sector.

The scope of the result is presently ADE. The same paper emphasizes that non-simply-laced cases introduce extra parameters and do not fit into the same refined universal formula, which aligns with the separate mixed-root-system theory described above (Bishler et al., 22 May 2025).

4. Superspace, pairs of diagrams, and multipartition generalizations

Macdonald polynomials in superspace provide another precise source of mixed dimension-like data. For a superpartition $R$1, the norm formula is

$R$2

where the product is over bosonic boxes $R$3, but the arm- and leg-lengths are computed in two different diagrams, $R$4 and $R$5 (González et al., 2018). The numerator uses arms from $R$6 and legs from $R$7, while the denominator uses arms from $R$8 and legs from $R$9. The same mixed arm/leg product reappears in the evaluation formulas $m_\lambda^R$0 and $m_\lambda^R$1, so the paper identifies the norms and evaluations as superspace analogues of Macdonald dimensional data (González et al., 2018).

The stable sector of Macdonald superpolynomials leads to double Macdonald polynomials indexed by pairs of partitions $m_\lambda^R$2. Their defining factorization is

$m_\lambda^R$3

and their double $m_\lambda^R$4-Kostka coefficients specialize at $m_\lambda^R$5 to dimensions of irreducible representations of the hyperoctahedral group $m_\lambda^R$6 (Blondeau-Fournier et al., 2012). This produces a genuine type-$m_\lambda^R$7 theory of mixed Macdonald dimensions.

An $m_\lambda^R$8-alphabet version, the multi-Macdonald polynomials $m_\lambda^R$9, is indexed by multipartitions $x=q^{2r_k^*}$00 and factors into a product of ordinary Macdonald polynomials with shifted parameter pairs $x=q^{2r_k^*}$01 evaluated on recursively defined alphabets (González et al., 2019). The associated multi $x=q^{2r_k^*}$02-Kostka coefficients are positive and satisfy

$x=q^{2r_k^*}$03

so they are $x=q^{2r_k^*}$04-analogues of dimensions of irreducible representations of $x=q^{2r_k^*}$05 (González et al., 2019).

A related, though more conjectural, diagrammatic notion appears for generalized Macdonald polynomials depending on two diagrams $x=q^{2r_k^*}$06 and two sets of times. On a codimension-one slice of the topological locus, their specialized value factorizes into three pieces: a Macdonald-dimension-like factor for $x=q^{2r_k^*}$07, another for $x=q^{2r_k^*}$08, and a genuinely mixed factor over pairs of boxes from $x=q^{2r_k^*}$09 and $x=q^{2r_k^*}$10 involving the parameter $x=q^{2r_k^*}$11 (Kononov et al., 2016). This suggests a two-diagram version of mixed Macdonald dimensions.

5. Probabilistic, half-space, and free-field realizations

Periodic Macdonald processes furnish a probabilistic realization of Macdonald dimensions. Under the Macdonald-Plancherel specialization $x=q^{2r_k^*}$12, the skew functions become

$x=q^{2r_k^*}$13

where $x=q^{2r_k^*}$14 and $x=q^{2r_k^*}$15 are path weights in the Young graph (Koshida, 2020). The shift-mixed periodic Macdonald measure extends this to $x=q^{2r_k^*}$16, with charge $x=q^{2r_k^*}$17 weighted by $x=q^{2r_k^*}$18; its partition function is

$x=q^{2r_k^*}$19

and charge-modified observables acquire a theta correction factor $x=q^{2r_k^*}$20 (Koshida, 2020). In that setting, mixed Macdonald dimensions are explicit weighted sums over partitions together with charge.

Half-space Macdonald theory yields a boundary-modified variant. The half-space measure on a single partition is

$x=q^{2r_k^*}$21

where $x=q^{2r_k^*}$22 is a Littlewood-type transform of $x=q^{2r_k^*}$23 involving an even-partition constraint (Barraquand et al., 2018). A natural definition suggested there is

$x=q^{2r_k^*}$24

with $x=q^{2r_k^*}$25 interpreted as a bulk contribution and $x=q^{2r_k^*}$26 as a boundary factor (Barraquand et al., 2018).

The free-field approach to the Macdonald process identifies the symmetric-function space with a Heisenberg Fock space and realizes Macdonald operators as contour integrals of vertex operators $x=q^{2r_k^*}$27 and $x=q^{2r_k^*}$28 (Koshida, 2019). The observables

$x=q^{2r_k^*}$29

and their $x=q^{2r_k^*}$30-inverted analogues are diagonal operator eigenvalues. Their expectations under Macdonald processes admit determinantal integral formulas and, in the one-step case, Fredholm determinant expressions (Koshida, 2019). In the Schur limit $x=q^{2r_k^*}$31, this determinantal structure becomes the ordinary free-fermion Wick determinant.

A still more elaborate mixed theory appears through generalized Macdonald functions arising from tensor products of Fock spaces and the Hopf algebra structure of the Ding–Iohara–Miki algebra. The generalized Macdonald measure on $x=q^{2r_k^*}$32,

$x=q^{2r_k^*}$33

packages multi-component Macdonald spectra, and the generalized observable $x=q^{2r_k^*}$34 has an explicit contour-integral expectation (Koshida, 2019). This is a direct multi-species realization of mixed Macdonald dimensions.

6. Operators, geometry, limits, and current status

The operator-theoretic underpinning for many mixed constructions is the generalized Macdonald difference operator $x=q^{2r_k^*}$35 attached to a small weight $x=q^{2r_k^*}$36 for an arbitrary admissible pair $x=q^{2r_k^*}$37. It acts diagonally on Macdonald polynomials,

$x=q^{2r_k^*}$38

and, by duality, yields explicit Pieri and Littlewood–Richardson type formulas when one factor is indexed by a small weight (Diejen et al., 2010). This suggests an operator-level interpretation of mixed Macdonald dimensions as $x=q^{2r_k^*}$39-deformed structure constants or multiplicities, although that terminology is interpretive rather than standard in the paper itself.

Geometric and cohomological interpretations enter through the Macdonald analogue of the Nekrasov–Okounkov hook-length formula. The identity

$x=q^{2r_k^*}$40

is a $x=q^{2r_k^*}$41-refinement of the original Nekrasov–Okounkov formula and is used to prove a main conjecture of Hausel and Rodriguez-Villegas on mixed Hodge polynomials of character varieties (Rains et al., 2016). In that setting, the hook-weighted partition sums become generating functions of mixed Hodge numbers, so the phrase mixed Macdonald dimensions acquires a literal cohomological content.

Several recurrent limits organize the subject. The unrefined limit $x=q^{2r_k^*}$42 sends Macdonald dimensions to quantum dimensions; the classical limit $x=q^{2r_k^*}$43 recovers ordinary dimensions; Hall–Littlewood, Jack, and Schur limits preserve the underlying mixed combinatorics while changing the ambient theory (Bishler, 15 Jul 2025). In superspace and double or multi theories, the same pattern persists: stable factorization survives, but the representation-theoretic interpretation shifts from type $x=q^{2r_k^*}$44 to type $x=q^{2r_k^*}$45, to wreath products, or to still hypothetical super-DAHA-type objects (Blondeau-Fournier et al., 2012).

At present, the term therefore has a layered meaning. Its most precise form is the mixed root-system evaluation $x=q^{2r_k^*}$46 for $x=q^{2r_k^*}$47 (Bishler, 15 Jul 2025). A second established form is the universal mixed sector $x=q^{2r_k^*}$48 in refined Vogel theory (Bishler et al., 22 May 2025). Beyond that, the literature supplies several coherent analogues—superspace norms and evaluations, double and multi $x=q^{2r_k^*}$49-Kostka coefficients, boundary-modified half-space weights, shift-mixed process partition functions, and generalized DIM-based measures—each encoding a refined dimension that mixes two structures rather than evaluating a single Macdonald object in isolation (González et al., 2018).