Promotion Maps: Theory and Applications

• Promotion maps are a heterogeneous family of transformations defined by local rules that yield global invariants in combinatorics, geometry, and economics.
• They employ mechanisms such as toggles, rowmotion, and pricing adjustments to facilitate optimization and maintain statistical properties.
• Applications range from Schützenberger promotion on tableaux and plabic tangle geometries to personalized promotion systems in large-scale platforms.

Searching arXiv for the cited papers and related promotion-map literature. Promotion maps are a heterogeneous family of maps whose meaning depends strongly on context. In algebraic combinatorics, the term typically refers to Schützenberger-type operators on tableaux and their piecewise-linear, birational, graph-theoretic, and Catalan analogues; in cluster and amplituhedron geometry, it denotes rational or algebraic maps between Grassmannians defined from plabic tangles; in platform economics and large-scale promotion systems, it denotes an explicit policy mapping observable variables such as prices, beliefs, histories, or user features to promotion probabilities or incentive levels (Johnson et al., 2022, Gur et al., 2019, Shen et al., 2021, Even-Zohar et al., 4 Aug 2025). Across these settings, promotion maps are dynamical or decision-theoretic transformations built from local rules, with emphasis on invariants, conjugacies, and optimization under constraints.

1. Classical origin and general definition

The classical source of the subject is Schützenberger promotion on tableaux. For a standard Young tableau $T$ of shape $\lambda$ with entries $1,\dots,n$, one deletes the entry $1$, performs jeu de taquin sliding until the skew tableau is rectified, subtracts $1$ from each remaining entry, and obtains a new tableau $\Pro(T)$. For semistandard tableaux, promotion is usually defined via Bender–Knuth involutions, and evacuation is the companion Schützenberger involution (Johnson et al., 2022).

A major modern formulation expresses promotion in terms of toggles on posets. For a finite poset $P$, the combinatorial toggle $t_p$ at $p\in P$ acts on order ideals by adding or removing $p$ when the result remains an order ideal; rowmotion is the product of toggles in any linear extension order. On the rectangle poset $\lambda$0, promotion on the Gelfand–Tsetlin region is realized as a product of rowmotion-restricted-to-a-file operators, and evacuation is a product of such promotions (Johnson et al., 2022). This toggle formalism is the basis for most later lifts and conjugacies.

The tableau state space has also been enlarged. A fluctuating tableau is a sequence

$\lambda$1

of generalized partitions in which each step adds or removes a skew column of size $\lambda$2. This class subsumes standard, transpose-semistandard, oscillating, vacillating, rational, and alternating tableaux. Promotion on fluctuating tableaux is defined via growth diagrams and local rules, and for rectangular fluctuating tableaux one has $\lambda$3 (Gaetz et al., 2023).

A comparable extension exists on graph labelings. If $\lambda$4 is a simple graph with $\lambda$5 vertices and $\lambda$6 is the set of bijective labelings $\lambda$7, then the toggle $\lambda$8 swaps labels $\lambda$9 and $1,\dots,n$0 when they lie on nonadjacent vertices and fixes the labeling otherwise. Classical-style promotion becomes

$1,\dots,n$1

while toric promotion is $1,\dots,n$2 (Defant et al., 2023). This replaces tableaux by graph labelings while preserving a toggle-group viewpoint.

2. Piecewise-linear and birational promotion on rectangles and moon polyominoes

A decisive development is the lift of toggle dynamics from combinatorial objects to polytopal and birational settings. On the order polytope $1,\dots,n$3, the piecewise-linear toggle at $1,\dots,n$4 is

$1,\dots,n$5

with empty $1,\dots,n$6 interpreted as $1,\dots,n$7 and empty $1,\dots,n$8 as $1,\dots,n$9. The same commutation relations as in the combinatorial setting hold, so rowmotion and promotion are defined by the same toggle products. Detropicalization gives the birational toggle

$1$0

with empty sum interpreted as $1$1 (Johnson et al., 2022).

This framework interacts closely with RSK. For $1$2 and $1$3,

$1$4

where $1$5 is Stanley’s transfer map from the order polytope to the chain polytope. The conjugated promotion map

$1$6

acts on real labelings of rectangles and controls Greene-type chain statistics

$1$7

defined as the maximum total weight of $1$8 nonintersecting NE-lattice paths in a subrectangle (Johnson et al., 2022).

The main structural result is a chain-shifting description of $1$9. For all $1$0 and all $1$1,

$1$2

while for all $1$3,

$1$4

Thus vertical Greene-type statistics are preserved and horizontal ones shift one step left (Johnson et al., 2022). Evacuation becomes rotation by $1$5 in RSK coordinates, and Striker–Williams promotion is conjugate to rowmotion.

These rectangle maps extend to moon polyominoes. A moon polyomino is convex and intersection-free, and equivalent moon polyominoes are related by permuting up-diagonals and down-diagonals. Rubey’s rectangle-wise bijections are lifted to piecewise-linear and birational maps $1$6 on fillings of equivalent moon polyominoes. These maps preserve the relevant chain statistics, commute when attached to different maximal rectangles, and imply Ehrhart equivalence: $1$7 They also imply Ehrhart quasi-polynomial period collapse for the associated clique-constraint stable set polytopes $1$8 (Johnson et al., 2022).

3. Graph-labeling, fluctuating-tableau, and rational Catalan extensions

On path graphs, promotion-type dynamics admit a complete orbit theory. For a bijection $1$9, permutoric promotion is

$\Pro(T)$0

If $\Pro(T)$1 is the number of cyclic descents of $\Pro(T)$2, then on $\Pro(T)$3 the order of $\Pro(T)$4 is $\Pro(T)$5, and the triple

$\Pro(T)$6

exhibits the cyclic sieving phenomenon (Defant et al., 2023). Broken promotion operators, gliding globs, sliding stones, and colliding coins provide the dynamical models used in the proof.

For fluctuating tableaux, promotion is not only a bijection on tableaux but also a source of finer invariants. To each rectangular fluctuating tableau $\Pro(T)$7, the paper associates a promotion matrix $\Pro(T)$8 and, for each $\Pro(T)$9, a promotion function $P$0 on the index set of individual box additions or removals. In the rectangular case each $P$1 is a permutation, one has

$P$2

and evacuation conjugates these permutations by the reverse permutation $P$3 (Gaetz et al., 2023). Antiexcedances of $P$4 recover row information: for standard rectangular tableaux, the antiexcedances of $P$5 are exactly the entries in the first $P$6 rows.

Rational Catalan combinatorics supplies another extension. On generalized Dyck paths, the four central maps are promotion $P$7, evacuation $P$8, rowmotion $P$9, and rowvacuation $t_p$0. A matching map $t_p$1 yields the conjugacy

$t_p$2

so promotion and rowmotion are equivalent dynamical systems on $t_p$3-Dyck paths (Shigechi, 18 Mar 2026). For classical Dyck paths, promotion corresponds to rotation on the noncrossing perfect matching, evacuation corresponds to the bar involution on labels, and rowvacuation is the Lalanne–Kreweras involution. For $t_p$4-Dyck paths, the same framework realizes rotation, Simion–Ullman, and Lalanne–Kreweras as compositions of promotion, evacuation, rowmotion, and rowvacuation (Shigechi, 18 Mar 2026).

4. Promotion maps in plabic geometry and cluster theory

A more recent meaning arises from plabic tangles. A plabic tangle consists of a plabic graph in an outer disk together with inner disks attached to internal black vertices. The construction uses $t_p$5-vector-relation configurations ($t_p$6-VRCs): vectors $t_p$7 on black vertices and nonzero scalars $t_p$8 on edges such that at every white vertex

$t_p$9

and the boundary vectors span $p\in P$0 (Even-Zohar et al., 4 Aug 2025).

For a reduced plabic graph $p\in P$1 of type $p\in P$2, the VRC fiber over a generic boundary configuration has cardinality equal to the $p\in P$3-intersection number $p\in P$4 when $p\in P$5. Hence $p\in P$6 is $p\in P$7-generically solvable if and only if $p\in P$8 and $p\in P$9 (Even-Zohar et al., 4 Aug 2025). Solvable, rank-$p$0 regular tangles therefore define geometric promotions

$p$1

and, after choosing a pinning, algebraic promotions

$p$2

The central conjecture states that for dominant solvable tangles there is a brushing and a choice of signs such that algebraic promotion restricts to a quasi-cluster homomorphism and geometric promotion preserves total positivity (Even-Zohar et al., 4 Aug 2025). This is proved for several families, including unary star promotion, BCFW promotion, unary spurion promotion, chain-tree promotion, and forest promotion.

The same paper identifies an operad structure on plabic tangles and on promotion maps. Composition of tangles induces composition of promotions, and dominant solvable tangles form a suboperad. The four-mass box example exhibits a different phenomenon: the associated promotion is algebraic rather than rational, has two branches

$p$3

and still sends every cluster variable of the source Grassmannian to a positive function on the positive Grassmannian (Even-Zohar et al., 4 Aug 2025). This suggests positivity properties beyond rational cluster maps.

5. Promotion maps as platform policies under learning and information design

In platform economics, a promotion map is an explicit stochastic policy. In the single-seller baseline, the platform observes a state $p$4, commits to a signaling mechanism $p$5 and a dynamic promotion policy $p$6, and chooses whether the seller is promoted through an action $p$7. The time-$p$8 promotion map is

$p$9

so it maps current price, underlying state, and seller-reconstructible history to a promotion probability (Gur et al., 2019).

Because the seller is Bayesian and learns $\lambda$00 from sales data, promotion maps affect not only current demand but also future pricing through learning. The paper introduces confounding promotion policies, defined so that at a fixed belief $\lambda$01 the seller’s belief never changes under myopic best responses. The confounding condition is

$\lambda$02

Under this equality, sale probabilities are identical across states, so sales are uninformative about $\lambda$03 (Gur et al., 2019).

The long-run design problem reduces to choosing a target price and state-dependent promotion probabilities. The paper shows that under myopic pricing one can restrict without loss to single-price promotion policies, and defines $\lambda$04 as the per-period consumer surplus attainable by a static confounding promotion map at belief $\lambda$05. The main asymptotic result is

$\lambda$06

the concavification of $\lambda$07 (Gur et al., 2019). A Bayesian Nash equilibrium is then constructed in which the seller prices myopically in every period, the platform uses a simple confounding promotion policy on path, and the resulting average consumer surplus equals the long-run optimum. The same profile is also horizon-maximin optimal (Gur et al., 2019).

6. Personalized promotion maps at massive scale

In large-scale promotion systems, promotion maps are individualized incentive-allocation rules. The framework in "A framework for massive scale personalized promotion" has two stages. Stage 1 learns user-level promotion-response curves

$\lambda$08

possibly together with resource predictors $\lambda$09. Stage 2 solves a constrained optimization problem whose decision variables are the assigned incentive levels. After discretizing the incentive space into $\lambda$10, the basic linear program is

$\lambda$11

subject to

$\lambda$12

Thus the promotion map is the optimizer $\lambda$13, or, operationally, the induced assignment $\lambda$14 (Shen et al., 2021).

The first stage must be counterfactual. Historical promotion data are treatment-biased, so the paper recommends inverse propensity score weighting and introduces the deep-isotonic-promotion-network (DIPN). DIPN combines a bias net, an uplift net, and an isotonic embedding

$\lambda$15

with nonnegative uplift weights $\lambda$16. This enforces that $\lambda$17 is non-decreasing in incentive. A smoothness penalty

$\lambda$18

is added to stabilize the learned response curves (Shen et al., 2021).

The resulting coefficients support large-scale optimization under multiple budgets or per-capita constraints. In online deployment, a shadow price $\lambda$19 for budget yields the per-user rule

$\lambda$20

which is the online promotion map given the current dual variable (Shen et al., 2021). Evaluation uses predictive metrics such as LogLoss and AUC-ROC, structural metrics such as RPR, EPR, and MLSS, and downstream metrics such as future response and future cost. In the reported production A/B tests for HuaBei campaigns, DIPN-based promotion maps reduced cost by $\lambda$21, $\lambda$22, and $\lambda$23, while usage rate changed by $\lambda$24, $\lambda$25, and $\lambda$26 across the three campaign types listed in the paper (Shen et al., 2021).

Promotion maps therefore constitute not a single theory but a family of technically precise constructions. In combinatorics they organize toggle dynamics, RSK, rowmotion, and evacuation on tableaux, graph labelings, Dyck paths, and polyomino fillings; in Grassmannian geometry they are rational or algebraic maps defined from plabic tangles and $\lambda$27-VRCs; in platform design they are policies or optimization outputs mapping observable states to promotion probabilities or incentive levels. A plausible implication is that the shared vocabulary reflects a common structural pattern: a local rule or policy specification is assembled into a global transformation whose significance lies in preserved statistics, conjugacy relations, or optimality properties.