$q$-Whittaker polynomials: bases, branching and direct limits

Aritra Bhattacharya; Sankaran Viswanath; T V Ratheesh

REVIEW 3 major objections 4 minor 1 cited by

$q$-Whittaker polynomials: bases, branching and direct limits

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Pith's one-line read This paper constructs two bijections between column strict fillings and partition overlaid patterns, making the CSF model carry the projection, branching, and direct-limit structure of local Weyl modules for the affine Lie algebra…

desk verdict Solid bijection paper with a load-bearing Proposition 9 that is asserted, not proved; the CSF character formula for L(Λ0) is conditional until that diagram is checked. read the letter →

arxiv 2412.00116 v1 pith:TNYQFIBC submitted 2024-11-28 math.CO math.RT

Aritra Bhattacharya , T V Ratheesh , Sankaran Viswanath This is my paper · ORCID

classification math.COmath.RT MSC 05E1005E05

keywords q-WhittakerpolynomialscolumnstrictfillingspartitionoverlaidpatternsinvstatisticquinvlocalWeylmodulesbasicrepresentationcolouredlatticepaths

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that column strict fillings, the natural indexing set for the inv and quinv expansions of the q-Whittaker polynomial, can be equipped with projection, branching, and direct-limit structures that exactly mirror those of partition overlaid patterns, the model tied to local Weyl modules of the affine Lie algebra $\widehat{\mathfrak{sl}}_n$. It constructs two explicit bijections $\psi_{\mathrm{inv}}$ and $\psi_{\mathrm{quinv}}$ between the two models that preserve monomials and $q$-weights, commute with the natural projection and branching maps, and differ by box complementation. If these bijections are correct, the finite CSF model carries the same module-theoretic information as the POP model, yielding CSF-native monomial bases of local Weyl modules and a new column-strict-filling formula for the character of the level-one vacuum module. A sympathetic reader would care because the paper turns a finite, row-sortable tableaux model into a vehicle for infinite-dimensional representation theory, and gives a concrete computable bridge between two previously separate combinatorial worlds.

What carries the argument

The load-bearing object is a pair of cellwise statistics on a column strict filling $F$: $\mathrm{zcount}(c,F)$, the number of quinv-triples whose third cell is $c$, and $\overline{\mathrm{zcount}}(c,F)$, the number of reflected inv-triples whose third cell is $c$. For $T=\mathrm{rsort}(F)$, both counts are bounded by $T^i_j-T^{i+1}_{j+1}$, and the two counts add to exactly this SE-difference, so reading either set of counts row by row yields the partition overlays that define a POP. The inverse bijections place entries one row at a time into labelled candidate cells. The branching structure is carried by the splice operation (a suffix swap between adjacent column tuples), iterated in the delete-and-splice algorithm, and the direct limit by the map that adds a prescribed pair of columns at each step.

What would settle it

For $n=3$ and $\lambda=\emptyset$, list all CSFs in $C_k$ for $k=0,1,2$ and compute the sum $\sum_{F\in C_k}x^F q^{k^2-\mathrm{inv}(F)}$; compare each monomial coefficient with the known $\theta$-function expansion of $\chi_{\Lambda_0}$. A single mismatch, or a failure of the fiber identity $\sum_{\mathrm{rsort}(F)=T}q^{\mathrm{inv}(F)}=\mathrm{wt}_q(T)$ on a small shape such as $(2,1)$, would overturn the paper's main structural claims.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: for every partition $\lambda$ with at most $n$ nonzero parts, there exist two bijections $\psi_{\mathrm{inv}},\psi_{\mathrm{quinv}}:\mathrm{CSF}(\lambda)\to\mathrm{POP}(\lambda)$ with the following properties: the monomial of $F$ equals the monomial of the GT pattern in its image, the statistic $\mathrm{inv}(F)$ (respectively $\mathrm{quinv}(F)$) equals the total size of the overlaid partitions, the projection $\mathrm{rsort}$ commutes with the POP projection, the branching map $\mathrm{dsplice}$ commutes with the POP branching map, and $\psi_{\mathrm{quinv}} = \mathrm{boxcomp}\circ\psi_{\mathrm{inv}}$. The proof machinery is cellwise: counting quinv-triples ending at a cell produces the quinv overlay, counting reflected inv-triples produces the complementary inv overlay, and the two counts always sum to the same SE-difference of the projected GT pattern. From this the paper obtains that the standard monomial basis of a local Weyl module can be indexed natively by CSFs with grades $\mathrm{inv}$ or $\mathrm{quinv}$, and that the direct limit of the CSF chain (append a column $2,3,\dots,n$ on the left and a column $1$ on the right) computes the character of the basic representation.

Load-bearing premise

The direct-limit character formula rests on Proposition 9, which asserts that the new CSF injection $s$ commutes with the previously defined POP injection $S$; $S$ is only cited from earlier work, not defined here, and the commutativity is stated as a 'simple consequence of the definitions' without proof, so if that diagram fails the limit formula does not follow.

Editorial extensions

If this is right

For each GT pattern $T$, the fibers of $\mathrm{rsort}$ have $q$-generating function $\mathrm{wt}_q(T)$, and $\mathrm{inv}+\mathrm{quinv}$ is constant on each fiber.
The involution $\Omega=\psi_{\mathrm{inv}}^{-1}\circ\psi_{\mathrm{quinv}}$ swaps $\mathrm{inv}$ and $\mathrm{quinv}$ while preserving the row-sorted tableau, giving an explicit bijection of the kind asked about in the quinv literature.
The sets $\{b_v(F)w_\lambda:F\in\mathrm{CSF}(\lambda)\}$, for $v=\mathrm{inv},\mathrm{quinv}$, are homogeneous monomial bases of the local Weyl module, with $q$-grade $v(F)$ and weight $x^F$.
The direct-limit chain yields $\chi_{\Lambda_0}=\sum_{k\ge0}\sum_{F\in C_k(\lambda)}x^F q^{\|\lambda+k\theta\|^2/2-\mathrm{inv}(F)}$, and for $\lambda=\emptyset$ the simpler form with $q^{k^2-\mathrm{inv}(F)}$.
In the coloured lattice path model, solid circles (intersections) read off the quinv overlay while open circles (non-intersections) read off the inv overlay, giving a simultaneous visual proof of both weight identities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Proposition 9 is supplied with a fully written proof, the CSF model likely becomes the most explicit route to affine Demazure characters, since its direct-limit map $s$ is described locally while the POP map $S$ is only cited.
The same cellwise counts may adapt to modified Hall-Littlewood polynomials and their quasisymmetric generalizations, yielding branching-friendly bases in those settings as well.
The intersection/non-intersection encoding suggests a purely path-theoretic proof of the fermionic formula that never mentions POPs; this is testable by checking whether the tile-by-tile counts satisfy the braid relations directly.
A computational check of the equivalences on small shapes would also test whether $\psi_{\mathrm{inv}}$ and $\psi_{\mathrm{quinv}}$ can be built recursively from elementary splices, which would give a simpler presentation of the bijections.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core of this paper is Theorem 2, the two bijections between column strict fillings and partition overlaid patterns, and it is mostly solid. The construction of ψinv and ψquinv via cellwise zcounts is genuinely new, the five-case analysis in Lemma 5 is real work, and the inverse algorithm in Section 9.2 gives an explicit way to check the bijections. The splice/dsplice branching formalism is also new, and Proposition 1 is proved carefully enough for the restricted case the paper needs. I found no specific error in the checked steps of the main theorem, and the lattice path readout in Section 11 is a nice visual, even if the verifications there are left to the reader.

The soft spot is exactly where the stress-test note points. Proposition 9 asserts that the CSF insertion map s is conjugated by ψinv to the map S from [RRV18], and the proof is one sentence: “a simple consequence of the definitions.” But S is not defined in this paper, and ψinv is a nontrivial bijection built over two sections. The direct-limit identification and the new character formula in Proposition 10 and Corollary 7 rest entirely on that diagram. As written, the L(Λ0) result is unverified — not necessarily wrong, but not established here. The authors also defer the full braid relations for the splice operators, but that is less serious: Corollary 2 proves exactly the two-length case needed for dsplice well-definedness.

Two smaller things. The dependence on [RRV18] is real but legitimate: it is a published construction, and the overlap is with one coauthor, not hidden. The paper is honest about what it defers — Proposition 9 is flagged only by its brevity, but the braid relations are explicitly stated as future work. The citation pattern looks fine.

Bottom line: this deserves a serious referee. The main bijection theorem should hold up, and the direct-limit application is worth checking. The referee needs to write out S, prove Proposition 9, or at least give a convincing reduction to [RRV18]. If that step closes, this is a strong paper. If it does not, the CSF character formula is only a conjecture and should be labeled as such.

Referee Report

3 major / 4 minor

Summary. The paper studies monomial expansions of q-Whittaker polynomials in the column-strict-filling (CSF) model and the partition-overlaid-pattern (POP) model. Its central result, Theorem 2, constructs two bijections psi_inv and psi_quinv from CSF(lambda) to POP(lambda) that preserve x-weight and the respective q-statistic, commute with the projection maps rsort/pr and the branching maps dsplice/br, and are related by box complementation. The proof introduces splice operations on columns, a delete-and-splice branching map, cellwise zcount statistics, and a refinv statistic. The paper also claims CSF-native Chari-Loktev bases (Proposition 2), a direct-limit construction for CSFs, and a new character formula for the basic representation L(Lambda_0) of the affine Lie algebra (Propositions 9-10, Corollary 7), and it closes with a lattice-path interpretation of the bijections.

Significance. If the main claims hold, this is a useful contribution: it gives an explicit, structure-preserving dictionary between two standard combinatorial models for q-Whittaker polynomials, resolves part of the Ayyer-Mandelshtam-Martin question on inv/quinv bijections, and provides a CSF-native perspective on Chari-Loktev bases and on direct limits towards the affine basic representation. The proof of Theorem 2 is detailed and supported by explicit constructions: Proposition 5 gives the clean complement relation between zcount and zcount, Lemma 5 is a five-case verification of splice-invariance, and Section 9.2 contains an explicit inverse algorithm. The advertised direct-limit character formula, however, rests on an unproved commutativity statement with an external map S, and Proposition 2 contains false statements as written. These issues are localized but must be repaired before the paper's full claims are acceptable.

major comments (3)

[§10.4, Proposition 9] Proposition 9 is load-bearing for the direct-limit identification and for the new character formula of Proposition 10 and Corollary 7, but it is not proved. The map S is not defined in the paper; the text refers the reader to [RRV18, §6], and the commutativity of the diagram with s and psi_inv is dismissed as 'a simple consequence of the definitions'. Since psi_inv is a nontrivial bijection constructed in Sections 8–9, this assertion cannot be checked without spelling out S and verifying the diagram. Please reproduce the definition of S (or make the relevant statement from [RRV18] self-contained) and give a proof of the commutativity. Until then, the direct-limit character formula should be regarded as unverified.
[§9.4, Proposition 2(3), Eqs. (47)–(48)] As written, Eqs. (47)–(48) are not a correct description of the Chari–Loktev monomials CL(P_v). The product ranges over all cells c in dg(lambda), but for a cell with F(c)=i(c) the symbol E_{F(c),i(c)} is not an element of n^-[t] (e.g., row-1 cells containing 1 would give E_{1,1}); only cells in cells(i,j,F) with 1 ≤ i ≤ j < n occur in CL(P_v). Moreover, zero zcount values are not removable: in CL(P), a part of size 0 contributes a factor E_{j+1,i} ⊗ 1. Accordingly, in the displayed example for F = 1 2 1 2 / 3 4, the cell (2,1) with entry 3 has zcount = 0 but contributes E_{3,2} ⊗ 1 to CL(P_quinv), and that factor is missing from the displayed b_quinv(F). The formula needs a restricted product over the cells of cells(i,j,F), 1 ≤ i ≤ j < n, with zero-exponent factors retained.
[§6, Proposition 2(2)] Proposition 2(2) is false as stated. Take n = 3, lambda = (3,2), F1 = (1 2 1 / 2 3) and F2 = (1 1 2 / 2 3). Both are column-strict and rsort(F1) = rsort(F2) with T = (1 1 2 / 2 3). For c = (1,2), F1(c) = 2 and the sum (zcount(c,F1) + zcount(c,F1)) equals 1 (the unique contributing triple is a refinv-triple with x = (1,3), y outside the diagram), whereas F2(c) = 1 makes both summands equal to 0. Hence the claimed equality fails for this pair. The statement should be corrected, for instance by restricting to cells with a fixed value F(c) = j+1, or removed if it is not needed.

minor comments (4)

[Throughout] The reference [RRV18] is typeset inconsistently as 'RR V18' or 'RRV18' in several places; this should be normalized.
[§8.1, proof of Proposition 4] The line 'X y∈Des(y) coarm(y↑)' contains a typo; it should be a sum over y ∈ Des(F).
[§9.4, Example 3] The displayed zcount values for F = 1 2 1 2 / 3 4 appear inconsistent with Definition 6: for the cell (1,3) of value 2, the triple with x = (1,1) and y = (2,1) is a quinv-triple (1 < 2 < 3), so its zcount is at least 1, while the displayed row has a 0 there.
[§10.4] In the definition of s(F), the text says 'for F in CSF(lambda + ktheta)' but then writes s(F) in CSF(lambda + (k+1)theta); this is clear from context, but the notation for the map's domain and codomain should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core bijections are constructed and proved from first principles, and the direct-limit reliance on [RRV18] is independent support; the main weak point is an unproved diagram, not a circular reduction.

full rationale

The central bijections ψ_quinv and ψ_inv are explicitly constructed from the rowsort T of a filling F and from cellwise statistics zcount / zcount (Sections 7–8). The equalities quinv(F)=|Λ| and inv(F)=|Λ| are proved by summing those cellwise counts, and pr(ψ_v(F))=rsort(F) holds by construction. The q-Whittaker expansions (3) and (5) are quoted from [HHL05] and [AMM23] as independent inputs, while the POP fermionic expansion is quoted from [Mac95] and [RRV18]; the bijections reconcile these known formulas rather than assume the target character formula. Branching compatibility in Theorem 2(2B) is proved by a splice case analysis and Tits' word reduction. The direct-limit section imports the map S and equation (72) from [RRV18]; despite one author overlap, [RRV18] is a published, parameter-free combinatorial construction with stated assumptions, so it is independent support, not a self-citation chain. Proposition 9 (Section 10.4) asserts commutativity of the CSF insertion s with the POP map S without defining S and with only the one-sentence justification that it is 'a simple consequence of the definitions.' That is an omitted proof and a genuine gap for the L(Λ0) character formula, but it is not a definitional circle: the assertion is not identical to its inputs by construction, and no fitted parameter is renamed as a prediction. No uniqueness theorem is imported from the authors' own prior work, and no known result is merely renamed. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted constants and no new postulated entities such as particles or forces. It relies on standard Macdonald/q-binomial facts and on prior representation-theoretic constructions (CL basis, local Weyl module character identity, S-map, direct limit to L(Λ0)), mainly from [CL06] and [RRV18]. [RRV18] is co-authored by the third author, so the POP machinery is self-cited; however, it is a published, parameter-free combinatorial construction with stated assumptions, so per the review rules it counts as independent support. The genuinely new content (splice/dsplice, the bijections, cellwise zcounts) is derived inside the paper.

assumptions (4)

domain assumption Wλ(Xn;q) equals the graded character of the local Weyl module Wloc(λ) of sl_n[t].
Invoked in the introduction and Section 4.4 via [CL06, Cor 1.5.2] and [FL07, Cor A]; the CL basis and branching arguments build on this identity.
domain assumption The Chari-Loktev monomials {CL(P)w_λ : P ∈ POP(λ)} form a homogeneous basis of Wloc(λ) with sl_n-weight x_T and q-grade |Λ|.
Theorem 1 is quoted from [CL06; RRV18] in Section 4.4; the CSF bases of Proposition 2 and Corollary 6 are defined by transferring this basis via the new bijections.
domain assumption There is a chain of injections Wloc(λ) ↪ Wloc(λ+θ) ↪ ... with direct limit L(Λ0), and an injective map S on POP with the properties listed in Proposition 8.
Quoted from [FL07] and [RRV18, §5-6] in Sections 10.2-10.3; the CSF direct limit and Corollary 7 depend on this machinery.
standard math The q-binomial coefficient generating function ∑_{γ⊆k×ℓ} q^{|γ|} = [k+ℓ choose k]_q.
Used in Section 4.1 to pass from the fermionic formula (18) to the POP expansion (22).

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Pith. "Pith review of $q$-Whittaker polynomials: bases, branching and direct limits." pith.science (2026). https://pith.science/paper/TNYQFIBC

@misc{pith2026241200116,
  author       = {Pith},
  title        = {Pith review of: $q$-Whittaker polynomials: bases, branching and direct limits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TNYQFIBC}},
  note         = {Machine review of arXiv:2412.00116}
}

abstract

We study $q$-Whittaker polynomials and their monomial expansions given by the fermionic formula, the inv statistic of Haglund-Haiman-Loehr and the quinv statistic of Ayyer-Mandelshtam-Martin. The combinatorial models underlying these expansions are partition overlaid patterns and column strict fillings. The former model is closely tied to representations of the affine Lie algebra $\widehat{\mathfrak{sl}_n}$ and admits projections, branching maps and direct limits that mirror these structures in the Chari-Loktev basis of local Weyl modules. We formulate novel versions of these notions in the column strict fillings model and establish their main properties. We construct weight-preserving bijections between the models which are compatible with projection, branching and direct limits. We also establish connections to the coloured lattice paths formalism for $q$-Whittaker polynomials due to Wheeler and collaborators.

Figures

Figures reproduced from arXiv: 2412.00116 by the authors.

**Figure 1.** A GT pattern for n = 4. The NE and SE differences are those along the red and blue lines. On the right is a partition overlay compatible with this GT pattern. where γ = (γ1 ≥ γ2 ≥ · · · ≥ γk ≥ 0) with ℓ ≥ γ1. Let Akℓ denote the set of such partitions γ. For later use, we also introduce the set Bkℓ comprising the strictly decreasing k-tuples a = (a1 > a2 > · · · > ak ≥ 0), with k + ℓ − 1 ≥ a1. One sees readily that A… view at source ↗

**Figure 2.** Box-complementation. Branching: Wλ(Xn; q) = X µ≺λ Y 1≤i<n λi − λi+1 λi − µi q Wµ(Xn−1; q) x |λ|−|µ| n (24) Both these also follow readily from (17) and (22). Definition 3. We define combinatorial projection to be the map pr : POP(λ) → GT(λ) pr(T,Λ) = T (25) and combinatorial branching to be the map br : POP(λ) → G µ≺λ POP(µ) br(T,Λ) = (T † ,Λ † ) (26) where the GT pattern T † is obtained from T ∈ GT(λ) by deleti… view at source ↗

**Figure 3.** An example of the rowsort operation. 5.1. Projection: rowsort. Given F ∈ CSF(λ), let rsort(F) denote the filling obtained from F by sorting entries of each row in ascending order from left to right ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: The following properties are immediate consequences of the definition: (i) The shape of Si(F) is si(dg(γ)) where si denotes the simple transposition in Sd which acts on diagrams of column compositions by swapping the columns i, i + 1. (ii) The multiset of entries of ea…

**Figure 5.** Figure 5: The dsplice operation intermixes columns. ≤ 5. The columns of F, F † , and dsplice(F) are colour-coded in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Decomposing a partition into rectangular sub-partitions. If j1, j2, · · · , jr is a sequence of j’s generated by iteration of step 2 of the algorithm, it follows from (45) that ji ∈/ D for all 1 ≤ i ≤ r. Thus, suppw γ ⊆ D c := {1 ≤ i < d : i /∈ D}, i.e., w γ ∈ Y , the …

**Figure 7.** Figure 7: Here F ∈ CSF(λ) for λ = (10, 6, 4, 0) and n = 4. Cells of F are coloured according to their entries. The gray cells are the extra cells in the augmented diagram dg( c λ). On the right are cellwise zcount values. Here quinv(F) = 12. 7.1. Cellwise zcounts. Given a partit…

**Figure 8.** Figure 8: (left to right) Configuration of quinv, inv and refinv triples. 7.3. Definition of ψquinv. We now have all the ingredients in place to define ψquinv. Let λ be a partition, F ∈ CSF(λ) and T = rsort(F). For each 1 ≤ i ≤ j + 1 ≤ n, consider the sequence Λij = (zcount(c, F…

**Figure 9.** Figure 9: A schematic diagram showing a column composition F † and the choice of αi , βi . α1, β2, γ and α2, β1, γ respectively. Let I1 = (α1, β1), I2 = (α2, β2), Ie1 = (α1, β2), Ie2 = (α2, β1) be the open intervals in Z with the indicated bounds. Observe that tk (resp. etk) is …

**Figure 10.** Figure 10: Here F ∈ CSF(λ) for λ = (4, 2, 0, 0) and n = 4. The cellwise zcount and zcount values are also indicated. Proof. To prove the forward implication: since F(z) = j +1, we have (i) F(x) ≤ j and (ii) F(y) > (j + 1). By (i), the cell x must have been empty at the stage whe…

**Figure 11.** Figure 11: Now let F ∈ CSF(λ). If F is just a single column, we represent it as a lattice path as follows: if the entries of F are i1 < i2 < · · · < ik, its lattice path: (i) starts at the right edge of the grid in row i1 and moves left or down (ii) has exactly k horizontal step…

**Figure 12.** Figure 12: Lattice path corresponding to a single-column CSF. I II III [PITH_FULL_IMAGE:figures/full_fig_p031_12.png]

**Figure 13.** Figure 13: A lattice path passing through a tile: possible configurations. of the paths. Let C, C ′ be columns of F and let γ, γ′ denote the lattice paths they correspond to. If the column C occurs to the left of C ′ in F, then the precedence rule for drawing the paths states th…

**Figure 14.** Figure 14: A CSF and its lattice path diagram (colour coded). Consider the tiles on the antidiagonal of the grid (shaded yellow in the example of [PITH_FULL_IMAGE:figures/full_fig_p031_14.png]

**Figure 15.** Figure 15: (a) The decluttering of [PITH_FULL_IMAGE:figures/full_fig_p033_15.png]

**Figure 16.** Figure 16: (a) T = rsort(F) for F in [PITH_FULL_IMAGE:figures/full_fig_p033_16.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Equating Inv-Quinv formulas for the $q$-Whittaker and modified Hall-Littlewood functions
math.CO 2024-12 accept novelty 4.0 of 10

The Inv and Quinv formulas for q-Whittaker and modified Hall-Littlewood functions are shown equal via the zeta and reversal maps on Carlsson-Mellit weighted Dyck paths.

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Works this paper leans on

12 extracted references · 8 canonical work pages · cited by 1 Pith paper

[1]

Modified Macdon- ald polynomials and the multispecies zero-range process. I

[AMM23] Arvind Ayyer, Olya Mandelshtam, and James B. Martin. “Modified Macdon- ald polynomials and the multispecies zero-range process. I”. In: Algebr. Comb. 6.1 (2023), pp. 243–284. [BB05] Anders Bj¨ orner and Francesco Brenti. Combinatorics of Coxeter groups. Vol

work page 2023
[7]

url: https://arxiv

arXiv: 2407.14099 [math.CO]. url: https://arxiv. org/abs/2407.14099. [Kac90] V. G. Kac. Infinite-dimensional Lie algebras . 3rd edition. Cambridge Univer- sity Press,

arXiv
[10]

Crystals for Demazure modules of classical affine Lie algebras

[Kun+98] Atsuo Kuniba et al. “Crystals for Demazure modules of classical affine Lie algebras”. In: J. Algebra 208.1 (1998), pp. 185–215. url: https://doi.org/ 10.1006/jabr.1998.7503. [Mac95] I. G. Macdonald. Symmetric functions and Hall polynomials . Second edition. Oxford Mathematical Monographs. Oxford University Press, New York, 1995, pp. x+475. REFERE...

work page arXiv 1998
[12]

q-Whittaker functions, finite fields, and Jordan forms

[KT21b] Steven N. Karp and Hugh Thomas. “ q-Whittaker functions, finite fields, and Jordan forms”. In: S´ emin. Lothar. Comb.85B (2021). Id/No 57, p

work page 2021
[13]

Percentage-avoiding, northwest shapes and peelable tableaux

arXiv: 2309 . 06401 [math.CO]. url: https : //arxiv.org/abs/2309.06401. [RS98] Victor Reiner and Mark Shimozono. “Percentage-avoiding, northwest shapes and peelable tableaux”. In: J. Combin. Theory Ser. A 82.1 (1998), pp. 1–73. url: https://doi.org/10.1006/jcta.1997.2841. [San00] Yasmine B. Sanderson. “On the connection between Macdonald polynomials and D...

arXiv 1998
[50]

Nonsymmetric Macdonald polynomials via integrable vertex models

url: https://doi.org/10. 1016/j.aim.2020.107449. [BW22] Alexei Borodin and Michael Wheeler. “Nonsymmetric Macdonald polynomials via integrable vertex models”. In: Trans. Amer. Math. Soc. 375.12 (2022), pp. 8353–8397. [CI15] Vyjayanthi Chari and Bogdan Ion. “BGG reciprocity for current algebras”. In: Compos. Math. 151.7 (2015), pp. 1265–1287. [CL06] Vyjaya...

work page arXiv 2022
[198]

Weyl modules, Demazure modules, KR-modules, crystals, fusion products, limit constructions

url: https://doi.org/10.1017/S0027763000026866. [FL07] G. Fourier and P. Littelmann. “Weyl modules, Demazure modules, KR-modules, crystals, fusion products, limit constructions”. In: Advances in Mathematics 211.2 (2007), pp. 566–593. [GW20] Alexandr Garbali and Michael Wheeler. “Modified Macdonald polynomials and integrability”. In: Commun. Math. Phys. 37...

work page doi:10.1017/s0027763000026866 2007
[231]

Springer, New York, 2005, pp

Graduate Texts in Mathematics. Springer, New York, 2005, pp. xiv+363. [Ber20] F. Bergeron. A Survey of q-Whittaker polynomials

work page 2005

Show all 12 references

[1990]

New combinatorial formula for modified Hall-Littlewood polynomials

[Kir00] Anatol N. Kirillov. “New combinatorial formula for modified Hall-Littlewood polynomials”. In: q-series from a contemporary perspective. American Mathe- matical Society, 2000, pp. 283–333. [KT21a] Steven N. Karp and Hugh Thomas. “ q-Whittaker functions, finite fields, a...

2021
[2020]

Monomial Expansions for q-Whittaker and Modified Hall-Littlewood Polynomials

arXiv: 2006.12591 [math.CO]. 34 REFERENCES [BR V24] Aritra Bhattacharya, TV Ratheesh, and Sankaran Viswanath. “Monomial Expansions for q-Whittaker and Modified Hall-Littlewood Polynomials”. In: S´ eminaire Lotharingien de Combinatoire, Issue 91B, Proceedings of the 36th Intern...

2006 arXiv
[2023]

Bijections between different combinatorial models for q-Whittaker and modified Hall-Littlewood polynomials

arXiv: 2309 . 16607 [math.CO]. url: https://arxiv.org/abs/2309.16607. [Rat24] TV Ratheesh. “Bijections between different combinatorial models for q-Whittaker and modified Hall-Littlewood polynomials”. In: Indian J Pure Appl Math (2024). url: https://doi.org/10.1007/s13226-024-...

2024 arXiv
[2024]

Spin q-Whittaker polynomials

[BW21] Alexei Borodin and Michael Wheeler. “Spin q-Whittaker polynomials”. In: Adv. Math. 376 (2021), Paper No. 107449,

2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Modified Macdon- ald polynomials and the multispecies zero-range process. I

[AMM23] Arvind Ayyer, Olya Mandelshtam, and James B. Martin. “Modified Macdon- ald polynomials and the multispecies zero-range process. I”. In: Algebr. Comb. 6.1 (2023), pp. 243–284. [BB05] Anders Bj¨ orner and Francesco Brenti. Combinatorics of Coxeter groups. Vol

work page 2023

[2] [7]

url: https://arxiv

arXiv: 2407.14099 [math.CO]. url: https://arxiv. org/abs/2407.14099. [Kac90] V. G. Kac. Infinite-dimensional Lie algebras . 3rd edition. Cambridge Univer- sity Press,

arXiv

[3] [10]

Crystals for Demazure modules of classical affine Lie algebras

[Kun+98] Atsuo Kuniba et al. “Crystals for Demazure modules of classical affine Lie algebras”. In: J. Algebra 208.1 (1998), pp. 185–215. url: https://doi.org/ 10.1006/jabr.1998.7503. [Mac95] I. G. Macdonald. Symmetric functions and Hall polynomials . Second edition. Oxford Mathematical Monographs. Oxford University Press, New York, 1995, pp. x+475. REFERE...

work page arXiv 1998

[4] [12]

q-Whittaker functions, finite fields, and Jordan forms

[KT21b] Steven N. Karp and Hugh Thomas. “ q-Whittaker functions, finite fields, and Jordan forms”. In: S´ emin. Lothar. Comb.85B (2021). Id/No 57, p

work page 2021

[5] [13]

Percentage-avoiding, northwest shapes and peelable tableaux

arXiv: 2309 . 06401 [math.CO]. url: https : //arxiv.org/abs/2309.06401. [RS98] Victor Reiner and Mark Shimozono. “Percentage-avoiding, northwest shapes and peelable tableaux”. In: J. Combin. Theory Ser. A 82.1 (1998), pp. 1–73. url: https://doi.org/10.1006/jcta.1997.2841. [San00] Yasmine B. Sanderson. “On the connection between Macdonald polynomials and D...

arXiv 1998

[6] [50]

Nonsymmetric Macdonald polynomials via integrable vertex models

url: https://doi.org/10. 1016/j.aim.2020.107449. [BW22] Alexei Borodin and Michael Wheeler. “Nonsymmetric Macdonald polynomials via integrable vertex models”. In: Trans. Amer. Math. Soc. 375.12 (2022), pp. 8353–8397. [CI15] Vyjayanthi Chari and Bogdan Ion. “BGG reciprocity for current algebras”. In: Compos. Math. 151.7 (2015), pp. 1265–1287. [CL06] Vyjaya...

work page arXiv 2022

[7] [198]

Weyl modules, Demazure modules, KR-modules, crystals, fusion products, limit constructions

url: https://doi.org/10.1017/S0027763000026866. [FL07] G. Fourier and P. Littelmann. “Weyl modules, Demazure modules, KR-modules, crystals, fusion products, limit constructions”. In: Advances in Mathematics 211.2 (2007), pp. 566–593. [GW20] Alexandr Garbali and Michael Wheeler. “Modified Macdonald polynomials and integrability”. In: Commun. Math. Phys. 37...

work page doi:10.1017/s0027763000026866 2007

[8] [231]

Springer, New York, 2005, pp

Graduate Texts in Mathematics. Springer, New York, 2005, pp. xiv+363. [Ber20] F. Bergeron. A Survey of q-Whittaker polynomials

work page 2005