Grounded Partitions in Affine Crystals

• Grounded partitions are colored partitions with a prescribed terminal part that encode the crystal path data and energy corrections of affine highest weight modules.
• They formalize a combinatorial model where difference conditions, induced by crystal energy functions, dictate the spacing between colored parts.
• Extensions to multi-grounded partitions accommodate periodic ground states, thereby generalizing character formulas and product identities in representation theory.

Grounded partitions are coloured partitions with a prescribed terminal ground part, introduced by Dousse and Konan and motivated by the theory of perfect crystals. Their central role is representation-theoretic: the crystal-path data of affine highest weight modules—specifically the crystal weights and the energy function on tensor products—can be encoded as explicit difference conditions on coloured integers, so that partition generating functions recover character formulas. The framework was first developed for constant ground state paths and then generalized to multi-grounded partitions for periodic ground state paths, thereby extending the model from a restricted class of modules to all perfect-crystal ground state paths of affine type (Dousse et al., 2021, Dousse et al., 2021, Dombos et al., 4 Aug 2025).

1. Crystal-path origin

The foundational setting is the path model for a level $\ell$ standard module $L(\lambda)$ of an affine Lie algebra. For a perfect crystal $B$, Kashiwara–Misra–Miwa–Nakashima theory realizes the crystal $B(\lambda)$ as a set of semi-infinite paths

$$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$

where $(g_k)_{k\ge0}$ is the ground state path of weight $\lambda$. The character is

$\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$

and the path weight incorporates both the crystal weights of the path components and an energy correction from adjacent tensor factors: $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$ The decisive combinatorial step is to reinterpret this path weight formula as a generating series for coloured partitions. In this reinterpretation, the colours record crystal elements, while the allowed differences between adjacent parts are determined by the energy function. Grounded partitions provide the model when the ground state path is constant; multi-grounded partitions provide the corresponding model when the ground state path is periodic (Dousse et al., 2021).

This origin fixes the conceptual status of grounded partitions. They are not an ad hoc variant of ordinary partitions with a distinguished smallest part. Rather, they are a crystal-theoretic encoding of semi-infinite paths, with the terminal ground data reflecting stabilization of the path and the difference conditions reflecting the normalized local energy.

2. Formal definition and combinatorial data

A generalised coloured partition is a finite sequence of coloured integers

$\pi=(\pi_0,\dots,\pi_s),$

where each $L(\lambda)$0 has a colour from a set $L(\lambda)$1, and the parts satisfy a prescribed binary relation $L(\lambda)$2. Its weight and colour word are

$L(\lambda)$3

A grounded partition is the special case in which the partition ends in a prescribed ground part. For a chosen ground colour $L(\lambda)$4, it is a nonempty generalised coloured partition

$L(\lambda)$5

subject to the chosen difference relation; in the simplest setting, the last part is $L(\lambda)$6, and the penultimate part must be different from it in the relevant colour sense (Dousse et al., 2021).

In the perfect-crystal setting, the binary relations are induced by the energy function. For a perfect crystal $L(\lambda)$7, one introduces a colour set $L(\lambda)$8, and after a suitable normalization the relations on coloured integers are

$L(\lambda)$9

and

$B$0

where $B$1 is an integer clearing denominators and $B$2 is a shifted energy function. The relation $B$3 gives exact difference conditions, while $B$4 gives weak difference conditions (Dousse et al., 2021, Dousse et al., 2021).

The formalism is flexible enough to distinguish several layers of structure at once: the colour alphabet, the adjacency relation on colours, the ground part, and the difference matrix determined by energy. This makes grounded partitions particularly suited to non-specialized character formulas, where colour monomials must be retained before principal specialization.

3. Multi-grounded partitions and periodic tails

The extension from grounded to multi-grounded partitions addresses the generic situation in which the ground state path is not constant but periodic. If the ground state path has period $B$5,

$B$6

then the ground is no longer a single terminal coloured integer. Instead, one fixes a block of $B$7 terminal coloured integers

$B$8

satisfying

$B$9

and the cyclic relation

$B(\lambda)$0

A multi-grounded partition is then a nonempty generalised coloured partition ending with exactly this fixed terminal block, with the additional condition that the final block is not simply a repetition of the ground itself (Dousse et al., 2021, Dousse et al., 2021).

The required tail is uniquely determined by the normalized energy of the periodic ground state path. The shifted energy is defined by

$B(\lambda)$1

so that the total energy over one period averages to zero. Proposition-level formulas in the general theory determine the values $B(\lambda)$2 from the periodic ground-state energies and ensure both the cyclic adjacency condition and the sum-zero condition (Dousse et al., 2021).

The principal character theorem of the framework states that the character of $B(\lambda)$3 becomes a generating series of multi-grounded partitions: $B(\lambda)$4 and, more generally,

$B(\lambda)$5

The second formula reflects a factorization into a strict multi-grounded component and an ordinary partition component with parts divisible by $B(\lambda)$6 (Dousse et al., 2021, Dousse et al., 2021).

This generalization is structurally important: grounded partitions are the $B(\lambda)$7 case of the multi-grounded theory, while periodic ground-state paths of period $B(\lambda)$8 and higher are handled directly, without passing to auxiliary crystals.

4. Higher-level $B(\lambda)$9 and absolute-value difference conditions

A particularly explicit realization occurs for the level $$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$0 perfect crystal $$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$1 of $$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$2. Its elements are $$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$3, with weights

$$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$4

and energy matrix

$$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$5

For the standard module $$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$6, where

$$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$7

the ground-state path alternates: $$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$8 The shifted energy is

$$\mathcal P(\lambda)=\{(p_k)_{k\ge 0} : p_k=g_k \text{ for } k\gg 0\},$$9

which can be rewritten as

$(g_k)_{k\ge0}$0

This reformulation is the source of the absolute-value difference conditions that distinguish the resulting partition identities (Dousse et al., 2021).

With $(g_k)_{k\ge0}$1 and

$(g_k)_{k\ge0}$2

the paper defines grounded coloured partition classes $(g_k)_{k\ge0}$3 by the exact relation

$(g_k)_{k\ge0}$4

equivalently

$(g_k)_{k\ge0}$5

where $(g_k)_{k\ge0}$6 is the colour index of $(g_k)_{k\ge0}$7. The weak class $(g_k)_{k\ge0}$8 is defined by

$(g_k)_{k\ge0}$9

For $\lambda$0, if $\lambda$1 and $\lambda$2 count partitions of weight $\lambda$3 in these two classes, then

$\lambda$4

and

$\lambda$5

These identities are described as companions to Andrews–Gordon and Meurman–Primc identities. Their novelty lies not in the modular product side, which matches known product expressions, but in the simplicity of the combinatorial model: a single absolute-value difference condition between adjacent coloured parts replaces the more intricate coupled inequalities of the Meurman–Primc setting (Dousse et al., 2021).

The same paper also gives non-specialized character formulas with manifestly positive coefficients for the three level-$\lambda$6 standard modules of $\lambda$7. For $\lambda$8, the ground-state path is constant and ordinary grounded partitions suffice; for $\lambda$9 and $\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$0, the ground-state paths have period $\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$1, so multi-grounded partitions and an even-parity projection operator are used (Dousse et al., 2021).

5. Level $\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$2 grounded partitions in type $\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$3

The level $\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$4 theory in type $\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$5 has a particularly rigid local combinatorics. The defining matrix is

$\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$6

and for $\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$7 one has

$\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$8

Partitions are written in weakly increasing order, with an initial dummy part $\ch L(\lambda)=\sum_{p\in\mathcal P(\lambda)} e^{\wt(p)},$9, and consecutive differences are given exactly by the corresponding matrix entries. Thus the local rules are: $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$0 gives difference $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$1, $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$2 gives $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$3, $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$4 gives $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$5; $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$6 gives $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$7, $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$8 gives $\wt(p)=\lambda + \sum_{k\ge0}(\wt p_k-\wt g_k) -\frac{\delta}{d_0}\sum_{k\ge0}(k+1)\bigl(H(p_{k+1}\otimes p_k)-H(g_{k+1}\otimes g_k)\bigr).$9, $\pi=(\pi_0,\dots,\pi_s),$0 gives $\pi=(\pi_0,\dots,\pi_s),$1; and $\pi=(\pi_0,\dots,\pi_s),$2 gives $\pi=(\pi_0,\dots,\pi_s),$3, $\pi=(\pi_0,\dots,\pi_s),$4 gives $\pi=(\pi_0,\dots,\pi_s),$5, $\pi=(\pi_0,\dots,\pi_s),$6 gives $\pi=(\pi_0,\dots,\pi_s),$7 (Dombos et al., 4 Aug 2025).

Two partition families are singled out. For $\pi=(\pi_0,\dots,\pi_s),$8, with ground $\pi=(\pi_0,\dots,\pi_s),$9,

• all even parts are colored $L(\lambda)$00,
• odd parts are colored $L(\lambda)$01 or $L(\lambda)$02,
• every odd size must appear at least once,
• once the first occurrence of an odd part is chosen, later equal-size odd parts alternate between $L(\lambda)$03 and $L(\lambda)$04.

For $L(\lambda)$05, with ground $L(\lambda)$06,

• all odd parts are colored $L(\lambda)$07,
• even parts are colored $L(\lambda)$08 or $L(\lambda)$09,
• every even size must appear at least once,
• later equal-size even parts alternate between $L(\lambda)$10 and $L(\lambda)$11 (Dombos et al., 4 Aug 2025).

The generating functions are explicit infinite products: $L(\lambda)$12 The 2025 paper provides the first bijective proof of these product formulas. For $L(\lambda)$13, the target family is overpartitions into odd parts; for $L(\lambda)$14, it is partitions with distinct even parts. In the first case, the construction separates a grounded partition into a minimal grounded partition and a collection of removable even parts, converts the minimal part to odd parts, and uses bars to encode whether first occurrences arose from $L(\lambda)$15- or $L(\lambda)$16-coloured parts. In the second case, the same three-step philosophy is adapted to the parity-reversed setting (Dombos et al., 4 Aug 2025).

These level-$L(\lambda)$17 results show that grounded partitions can support both crystal-theoretic and explicitly bijective partition-theoretic analyses. The product expressions are therefore not merely character-theoretic consequences; they admit direct combinatorial realizations.

6. Affine crystal structure, finite-type restriction, and significance

Grounded partitions at level $L(\lambda)$18 in type $L(\lambda)$19 carry an affine crystal structure. The paper constructs this structure on $L(\lambda)$20, realizing the crystal of highest weight

$L(\lambda)$21

The cells are labeled $L(\lambda)$22, with $L(\lambda)$23-cells green, $L(\lambda)$24-cells blue, and $L(\lambda)$25-cells inheriting color from neighboring data in the row; crystal arrows are defined by bracketing addable and removable corners of each color. The resulting crystal is identified, via an explicit bijection $L(\lambda)$26, with the Jimbo–Misra–Miwa–Okado pair-of-partitions model (Dombos et al., 4 Aug 2025).

Restricting this affine crystal to the finite type $L(\lambda)$27 crystal structure yields new infinite sum expressions for the product formulas. A central combinatorial fact is that a grounded partition is a starting point of a blue string if and only if the sequence of $L(\lambda)$28-labels, read from smallest nonzero part upward, forms a Yamanouchi word under the identification

$L(\lambda)$29

The major index enters through a modified statistic on Yamanouchi words, and the level-$L(\lambda)$30 product becomes

$L(\lambda)$31

The appearance of the modified major index is described as natural within this decomposition, rather than as an external statistic superimposed on the partition model (Dombos et al., 4 Aug 2025).

Taken together, the general character formulas, the higher-level $L(\lambda)$32 absolute-value identities, and the level-$L(\lambda)$33 bijective and crystal-theoretic refinements determine the present scope of grounded partitions. They function simultaneously as:

• a partition model for crystal paths in perfect crystals,
• a mechanism for positive, often non-specialized character formulas of affine Kac–Moody modules,
• a source of Rogers–Ramanujan–Andrews–Gordon-type product identities with unusually simple coloured difference conditions,
• and, at least in type $L(\lambda)$34 level $L(\lambda)$35, an affine-crystal object with explicit bijections to classical partition families (Dousse et al., 2021, Dousse et al., 2021, Dombos et al., 4 Aug 2025).

A recurrent structural point is that the “ground” is not arbitrary. In grounded partitions it is the fixed terminal part $L(\lambda)$36; in multi-grounded partitions it is a uniquely determined periodic tail. This suggests that the defining feature of the theory is not merely colouring or spacing, but the encoding of asymptotic crystal behavior by a rigid terminal datum.