{"id":"0300e8e5-49ed-4265-b3c0-2881756eae00","arxiv_id":"2412.09929","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This note explains why two different combinatorial formulas for the same q-Whittaker and Hall-Littlewood symmetric functions agree. It does so by encoding both formula sets as weighted statistics on Dyck paths and showing the two paths are related by known transformations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the most delicate step is Lemma 2.7, but its splice arithmetic and diagonal-preservation claim check out on inspection, so the path-transformation proof of Theorem 2.8 stands.","rationale":"The reader correctly identifies Lemma 2.7 as the most delicate step: it is a hand-checked splice case analysis and is not machine-verified. My stress-test focused exactly there. Recomputing the two-block splice cases (both m=0 and m>0 for decreasing words, and the corresponding weakly-increasing argument) reproduces the claimed one-extra-dinv and zero-extra-dinv counts. The assertion that the splice preserves each entry's diagonal is the key reduction allowing the two-column analysis to suffice for arbitrary numbers of blocks; geometric bookkeeping of the reading labels in a swapped two-block subpath supports this assertion. The final q-exponent in Theorem 2.8 matches the alpha difference in Lemma 2.2, and the high-degree equality has no q-correction, consistent with the second half of Lemma 2.7. Because the theorem's conclusion is already implied by known results, the main contribution is the explanatory path-transformation proof, and I found no gap in that explanation. A computational verification of Lemma 2.7 would still be a useful independent check, but I do not see a load-bearing concern that should change the reader's acceptance.","tokens_in":11794,"tokens_out":29824,"duration_ms":320043,"concrete_test":"Run an independent brute-force verification of Lemma 2.7 for all balanced paths N^{\\ell_1}E^{\\ell_1}\\cdots N^{\\ell_n}E^{\\ell_n} with total semilength at most 10 and all adjacent swaps with \\ell_i<\\ell_{i+1}, comparing \\bar\\chi(\\pi',q,0)=q\\,\\bar\\chi(\\pi,q,0) and equality of the highest t-degree coefficients coefficient-wise in x. This would confirm the q-factor and the high-degree invariance without relying on the hand-checked splice analysis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The load-bearing step is Lemma 2.7, which computes the change in the lowest and highest t-degree terms of the weighted characteristic function under adjacent swaps of bounce blocks. The proof reduces to a two-block splice and asserts that the splice does not change which diagonal any entry belongs to, so only dinv contributions between the two spliced columns change. I checked the two cases m=0 and m>0 in the decreasing-word part: the new dinv count is exactly one larger in both cases, and the weakly-increasing-word part preserves the dinv count. The diagonal-preservation claim is also geometrically sound: in a two-block subpath N^a E^a N^b E^b swapped to N^b E^b N^a E^a, the splice moves entries along their original diagonal classes, so interactions with other blocks are unchanged. The subsequent application of Lemma 2.7 in Theorem 2.8, with the q-exponent matching Lemma 2.2, is consistent. I did not find an internal inconsistency or a hidden assumption that would break the central chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves an equality of two known combinatorial formulas for q-Whittaker functions and modified Hall-Littlewood functions. The author encodes the HHL Inv formulas and the AMM Quinv formulas as weighted characteristic functions of two explicit Dyck paths, and then shows in Theorem 2.8 that the lowest t-degree terms of the two weighted path functions agree up to a q-power while the highest t-degree terms agree exactly. The proof connects the paths through the maps ζ, rev, and adjacent swaps of balanced blocks, relying on Proposition 2.4, Lemma 2.6, and Lemma 2.7. A final section contains remarks and examples on Schur positivity.","tokens_in":11952,"tokens_out":17495,"duration_ms":174146,"significance":"The result gives a direct and explicit combinatorial explanation for the equality of the Inv and Quinv formulas, rather than an indirect argument through the Macdonald polynomials. The transformation chain is explicit enough to serve as a proof of equality of the right-hand sides of (2.21) and (2.23). The paper is concise and mostly self-contained, and the delicate counting step in Lemma 2.7 checks out on inspection; the splice arithmetic and diagonal-preservation claim are consistent. The result will be useful to readers working on q-Whittaker and Hall-Littlewood combinatorics and on path models for symmetric functions.","major_comments":[],"minor_comments":[{"comment":"The expression \"qqstat(π,T)ttstat(π,T)sshape(T)\" appears to be a typesetting error for q^{qstat(π,T)} t^{tstat(π,T)} s_{shape(T)}; please fix it.","section":"Section 3, unnumbered display before (3.1)"},{"comment":"There is a stray extra period in the displayed line ending with \"(2.25). . Then\"; remove it.","section":"Section 2, after (2.25)"},{"comment":"The convention discussion for \\tilde H_λ versus \\tilde H_{λ'} is easy to misread; state explicitly that in (2.21) the modified Hall-Littlewood function is \\tilde H_{λ'}, while W_λ is the q-Whittaker function under the convention of (2.20).","section":"Section 2, (2.20)-(2.21)"},{"comment":"The sentence \"splice does not change which diagonal a particular entry belongs to\" is the reduction step that confines the dinv count to the two spliced columns; a short coordinate calculation or a sentence explaining why position indices determine diagonal classes would make this rigorous and easier to check.","section":"Lemma 2.7"}],"recommendation":"minor_revision","confidential_remarks":"No concerns for the editor. The speculative Schur-positivity remarks in Section 3 are clearly framed as remarks and do not affect the main theorem. The citation of the overlapping author's alternate proof [BRV24] is appropriate and not problematic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis note does what it says: it explains the known equality between the HHL Inv and AMM Quinv formulas by realizing both as lowest/highest t-degree terms of Carlsson-Mellit weighted path symmetric functions. The equality itself was already a corollary of the two formula theorems, and the paper is upfront about that; the contribution is the explanation. Proposition 2.4 and Corollary 2.5, which connect the two paths through zeta, reversal, and a balanced intermediate path, are genuinely nice. Lemma 2.7, the splice lemma, is a useful working tool, and the proof of Theorem 2.8 is coherent: the q-power shift in Lemma 2.2 matches the number of block swaps in Lemma 2.3, and the second statement goes through with no change in dinv.\n\nI checked the delicate part, Lemma 2.7, more carefully than the author's \"it suffices to consider two blocks\" handwave. The splice operation does preserve diagonal classes, so the only changed dinv contributions are between the two spliced columns. In the decreasing-word case, both m=0 and m>0 give exactly one extra dinv; in the weakly increasing case the new and old dinv contributions cancel. I did not find a gap. The citation to [BRV24] for the splice map is fine; it is an overlapping-author paper, but it is cited openly and the present lemma is proved directly.\n\nThe real soft spots are presentation, not math. The proof switches between the plain weighted characteristic chi(pi,q,t) and the composition chi(pi,q,t)=chi(zeta(pi),q,t) with notation that is easy to lose; in the text I saw, the underlined symbol looks identical to the plain one. Restore the underlines or introduce different letters, because the argument depends on keeping them separate. Lemma 2.7 would also benefit from a one-paragraph statement of why the splice map is a bijection on the relevant word sets; the case analysis computes dinv differences, but the map itself is imported informally. Section 3 is a remark and should be read as such; the Schur-positivity question and the computer check are nice conversation, not a claim.\n\nBottom line: this is a solid short note for algebraic combinatorics. It deserves a normal referee, and I would be happy to see it published as an explanatory note. Send it out.","headline":"A clean, honest path-transformation proof that the Inv and Quinv formulas agree; the equality was already known as a corollary, but the explanation is new and the proof checks out.","tokens_in":12525,"tokens_out":6321,"would_cite":true,"duration_ms":61017,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Inv and Quinv formulas for q-Whittaker and modified Hall-Littlewood functions are the same weighted Dyck path in different coordinates.","keywords":["q-Whittaker functions","modified Hall-Littlewood functions","Dyck paths","zeta map","Inv statistic","Quinv statistic","weighted path symmetric functions","Macdonald polynomials"],"falsifier":"For a partition with a repeated part size, say $\\lambda=(2,2,1)$, enumerate the words contributing to $\\chi(\\pi^{\\mathrm{Inv}}_\\lambda,q,0)$ and $\\chi(\\pi^{\\mathrm{Quinv}}_\\lambda,q,0)$ and check directly whether $\\chi(\\pi^{\\mathrm{Quinv}}_\\lambda,q,0)=q^{\\alpha_{\\mathrm{Quinv}}-\\alpha_{\\mathrm{Inv}}}\\chi(\\pi^{\\mathrm{Inv}}_\\lambda,q,0)$; any mismatch in a q-exponent would falsify the theorem.","tokens_in":11551,"feed_emoji":"🧮","tokens_out":11213,"duration_ms":100041,"temperature":0.7,"pith_summary":"The paper establishes that the two standard combinatorial formulas for the q-Whittaker and modified Hall-Littlewood functions—the Inv formula and the Quinv formula—are not merely both correct: they are evaluations of the same weighted Dyck path symmetric function on two paths connected by an explicit sequence of zeta and reversal maps. This explains the equality without matching individual word statistics case by case. The proof applies to both extremes of the t-expansion simultaneously, so the q-Whittaker identity and the Hall-Littlewood identity are obtained from one path-level calculation.","feed_headline":"Quinv equals Inv for q-Whittaker and Hall-Littlewood functions","feed_subtitle":"A zeta-and-reversal path transformation explains why the two formula families agree.","key_machinery":"The central object is the weighted path symmetric function $\\chi(\\pi,q,t)=\\sum_w q^{\\mathrm{inv}(\\pi,w)}t^{\\#\\{(i,j)\\in c(\\pi):w_i\\le w_j\\}}x^w$, a sum over positive integer words attached to a Dyck path. The two paths $\\pi^{\\mathrm{Inv}}_\\lambda$ and $\\pi^{\\mathrm{Quinv}}_\\lambda$ are built from the inversion and quinversion reading orders of $\\lambda$. The proof is carried by the zeta map $\\zeta$, the reversal map $\\mathrm{rev}$, and Lemma 2.7, which controls how $\\chi$ changes when adjacent bounce blocks of increasing length are swapped in a balanced Dyck path.","core_discovery":"The central claim is Theorem 2.8: for any partition $\\lambda$, the Quinv and Inv path evaluations agree after the same normalization, namely $q^{-\\alpha_{\\mathrm{Quinv}}(\\lambda)}\\chi(\\pi^{\\mathrm{Quinv}}_\\lambda,q,0)=q^{-\\alpha_{\\mathrm{Inv}}(\\lambda)}\\chi(\\pi^{\\mathrm{Inv}}_\\lambda,q,0)$, and at the top t-degree $\\chi(\\pi^{\\mathrm{Inv}}_\\lambda,q,t)|_{t^{\\#c}}=\\chi(\\pi^{\\mathrm{Quinv}}_\\lambda,q,t)|_{t^{\\#c}}$, with $\\#c=|\\lambda|-\\lambda_1$. The proof expresses $\\pi^{\\mathrm{Quinv}}_\\lambda$ as $\\mathrm{rev}\\circ\\zeta\\circ\\mathrm{rev}\\circ\\zeta^{-1}\\circ\\mathrm{rev}(\\pi^{\\mathrm{Inv}}_\\lambda)$ and then shows that each map changes the weighted path function in a controlled way.","pith_inferences":["The same zeta-reversal relation may connect the full t-deformations $\\chi(\\pi^{\\mathrm{Inv}}_\\lambda,q,t)$ and $\\chi(\\pi^{\\mathrm{Quinv}}_\\lambda,q,t)$, not just their lowest and highest t-degree terms; this is directly testable by computing both finite sums for small partitions.","The splice operation in Lemma 2.7 isolates the whole difficulty in a two-block swap, so the method should extend to other pairs of balanced Dyck paths that differ by adjacent swaps of increasing block lengths.","Since $\\chi(\\pi,q,0)$ and $\\chi(\\pi,q,1)$ are LLT polynomials, the path-level equivalence hints at a hidden relation between the LLT polynomials attached to $\\pi^{\\mathrm{Inv}}_\\lambda$ and $\\pi^{\\mathrm{Quinv}}_\\lambda$; a Schur-positive insertion statistic for intermediate $t$ would generalize the open question raised in Section 3."],"forward_implications":["The two formula families are the same sum written in different reading orders: both Inv and Quinv statistics are instances of the single inv statistic on Dyck paths.","The normalization difference $\\alpha_{\\mathrm{Quinv}}(\\lambda)-\\alpha_{\\mathrm{Inv}}(\\lambda)$ equals the length of the shortest permutation that reverses $\\lambda'$, so the q-power shift in the theorem has a concrete combinatorial meaning.","Because both the q-Whittaker and the modified Hall-Littlewood identities are read off from the same path function, proving one path-level relation proves both classical identities at once.","The path transformations are reversible, so the equality can be read in either direction between the Inv and Quinv models."],"supporting_citations":[{"why":"Supplies the Inv formula for q-Whittaker and modified Hall-Littlewood functions that Theorem 2.8 equates to the Quinv side.","marker":"[HHL05]"},{"why":"Supplies the Quinv formula for the same two symmetric functions, the other side of the equality.","marker":"[AMM23]"},{"why":"Defines the weighted characteristic function and the zeta-map transformation rules used throughout the proof.","marker":"[CM18]"},{"why":"Provides the definition and properties of the zeta map, including the corner description used to relate the two paths.","marker":"[HX17]"},{"why":"Gives an alternate proof of the same equality, cited as independent confirmation in the introduction.","marker":"[BRV24]"}],"fun_headline_variants":["Inv equals Quinv via a zeta-and-reversal path","Path transformation equates two symmetric function formulas","Why Quinv matches Inv: a path symmetry proof","New path map links q-Whittaker and Hall-Littlewood","Zeta-reversal unifies Inv and Quinv formulas"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is Lemma 2.7: swapping two adjacent bounce blocks of increasing length changes the lowest t-degree weighted sum by exactly one power of q and leaves the highest t-degree sum unchanged, and if that dinv count is off by even one unit, the chain of equalities in Theorem 2.8 breaks.","fun_headline_variants_meta":{"raw":{"variants":["Inv equals Quinv via a zeta-and-reversal path","Path transformation equates two symmetric function formulas","Why Quinv matches Inv: a path symmetry proof","New path map links q-Whittaker and Hall-Littlewood","Zeta-reversal unifies Inv and Quinv formulas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00064,"raw_usage":{"total_tokens":2869,"prompt_tokens":792,"completion_tokens":2077,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":408,"completion_tokens_details":{"reasoning_tokens":1996}},"tokens_in":408,"tokens_out":2077,"duration_ms":16167,"temperature":1.0,"reasoning_tokens":1996,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:33:16.221534+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a partition with a repeated part size, say $\\lambda=(2,2,1)$, enumerate the words contributing to $\\chi(\\pi^{\\mathrm{Inv}}_\\lambda,q,0)$ and $\\chi(\\pi^{\\mathrm{Quinv}}_\\lambda,q,0)$ and check directly whether $\\chi(\\pi^{\\mathrm{Quinv}}_\\lambda,q,0)=q^{\\alpha_{\\mathrm{Quinv}}-\\alpha_{\\mathrm{Inv}}}\\chi(\\pi^{\\mathrm{Inv}}_\\lambda,q,0)$; any mismatch in a q-exponent would falsify the theorem.","supporting_citations":[],"review_version":1}