{"id":"9da1b0d0-b55e-49e3-b354-579b547fe238","arxiv_id":"2505.19072","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hybrid Grothendieck polynomials unify stable and dual stable Grothendieck polynomials, with proofs of symmetry, Schur expansion, saturated Newton polytopes, and an omega image formula.","lead":"This paper introduces hybrid Grothendieck polynomials, a common generalization of stable and dual stable Grothendieck polynomials, and proves symmetry, a crystal-based Schur expansion, and saturated Newton polytopes for straight shapes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The crystal underpinning Theorem 1.3 depends on a braid relation in Lemma 2.9 that is asserted with 'it is easy to check' and not verified; this is the main load-bearing soft spot.","rationale":"The reader's weakest_assumption identifies the same foundational point: confluence of the descent-resolution algorithm on benign 12-SV tables. I agree that this is the most load-bearing assumption because it is used not only for the symmetry involution Φ of Theorem 2.2 but also, via norm(T), for the crystal operators e_i and f_i in Section 3.2 and therefore for the entire Schur expansion of Theorem 1.3. The paper gives a real proof structure, including the decreasing measure ℓ(T) and the reduction of confluence to Lemma 2.9, but the critical local braid identity in Case 2 is left as 'it is easy to check'. That is acceptable in principle, but it is exactly the kind of compressed verification that should be checked carefully, especially since the starred boxes in Figure 4 introduce set-valued cases beyond the known non-set-valued result of [20]. I do not see a more serious or more specific gap: the reconstruction argument in Proposition 3.12 is concrete, the embedding into word crystals in Theorem 3.13 is justified by the commutation of crystal operators with the reading word, and the positivity/SNP arguments in Section 4 follow from the Schur expansion once it is established. The typo in Conjecture 6.9 does not affect the main results. A conditional verdict is appropriate: the central combinatorial machinery is elaborate and mostly detailed, but the one asserted braid check should be independently verified before full acceptance. Hence I recommend no change to the reader's CONDITIONAL verdict.","tokens_in":34113,"tokens_out":10825,"duration_ms":104552,"concrete_test":"Enumerate exhaustively all local benign configurations of three adjacent columns relevant to Lemma 2.9 Case 2: column types 1-pure, 2-pure, and mixed with and without a {1,2} box, row heights up to 3, seplist satisfying Definition 2.4, and descents at columns j−1 and j. For each configuration, compute both sides of the claimed braid identity res_{j−1}(res_j(res_{j−1}(T))) = res_j(res_{j−1}(res_j(T))) using the rules in Figure 4, including both starred variants. Also verify Proposition 2.6(2) for the same configurations. If every instance passes, the confluence step is confirmed; any counterexample would invalidate Theorem 2.2 and hence Theorem 1.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3 requires SVRPP_n(λ/μ) to be a normal gl_n-crystal. The construction of the crystal operators in Section 3.2 calls on norm(T), which is defined through the descent-resolution algorithm of Section 2. The well-definedness of that algorithm is Proposition 2.10, proved by induction on ℓ(T). Its induction step uses Lemma 2.9, the local confluence statement. In the difficult Case 2 (i = j−1), the proof asserts the braid identity res_{j−1}(res_j(res_{j−1}(T))) = res_j(res_{j−1}(res_j(T))) with only 'it is easy to check', and no explicit case analysis is given for the set-valued variants: the two starred entries in (M1) and (2M) can each be either {2} or {1,2}, and benignity imposes subtle constraints on the separator list. This local relation is what forces the one-element normal form in Proposition 2.10, hence it makes Φ in Theorem 2.2 well-defined and makes e_i/f_i legitimate operators. If the braid relation failed for some benign configuration, the crystal graph would not be well-defined and the Schur expansion in Theorem 1.3 would not follow. The statement is plausible and likely routine, but it is load-bearing and currently asserted rather than demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines hybrid Grothendieck polynomials G_{λ/μ}(x;t;w) as generating functions over set-valued reverse plane partitions (SVRPPs) of skew shape λ/μ, with statistics ircont, ceq, and ex. It specializes to refined stable Grothendieck polynomials (t=0) and refined dual stable Grothendieck polynomials (w=0), and to Schur functions when both t=w=0. The main results are: (1) symmetry in x, proved by an explicit involution constructed from a flip and descent-resolution algorithm on benign 12-SV tables; (2) a Schur expansion of G_{λ/μ}(x_n;t;w) with coefficients counted by SVRPPs whose reading word is a reverse lattice word, obtained by building a gl_n-crystal on SVRPP_n(λ/μ); (3) the saturated Newton polytope (SNP) property for straight shapes, via a characterization of possible weights of highest weight elements; and (4) a combinatorial formula for the omega-involution image in the specialization t_i=α, w_i=β, derived through Fomin–Greene noncommutative Schur operators. The paper also formulates several open problems and conjectures.","tokens_in":34346,"tokens_out":9768,"duration_ms":87967,"significance":"If the technical gaps are repaired, this paper would be a substantial unification: it places refined stable and refined dual stable Grothendieck polynomials into one crystal-theoretic framework and recovers or extends results of Monical–Pechenik–Scrimshaw, Galashin, Chan–Pflueger, and Galashin–Grinberg–Liu. The SNP result for straight shapes and the omega formula for the two-parameter specialization are new and likely to be useful. The paper is clearly written, with many worked examples and an appendix listing the full crystal graph for a small shape, which strengthens its accessibility. The main caveats are that several load-bearing local statements are asserted without detailed proof (the braid relation in Lemma 2.9, properties of L_i in Lemma 4.5, and the correctness of the reconstruction algorithm in Proposition 3.12), and the definitional framework for subtableaux T(i) with empty boxes needs to be made precise before the crystal operators are fully well-defined.","major_comments":[{"comment":"The braid identity res_{j−1}(res_j(res_{j−1}(T))) = res_j(res_{j−1}(res_j(T))) is asserted with only 'it is easy to check'. Because Proposition 2.10 (the uniqueness of the normal form) depends on this identity, and hence so do the well-definedness of the involution Φ in Theorem 2.2 and the crystal operators e_i/f_i in Section 3.2, a complete verification is required. The proof must cover all set-valued variants (the two starred entries in (M1) taking values {2} or {1,2}, and the two starred entries in (2M) taking values {1} or {1,2}) together with the benignity constraints on seplist.","section":"§2.4, Lemma 2.9, Case 2"},{"comment":"The 'extra constraint' in Definition 2.4 refers to 'there exists a (unique) box in column k_{j+1} filled with the set {1,2}', but a mixed column in a benign 12-SV table can contain more than one box filled with {1,2} (for example, the column {1}, {1,2}, {1,2}, {2} is weakly increasing). The uniqueness assertion is therefore not justified, and the definition is ambiguous. The authors should specify which box is meant (e.g., the topmost such box) and then re-verify the descent-resolution lemmas under that clarification.","section":"§2.2, Definition 2.4"},{"comment":"The subtableau T(i) of a SVRPP T obtained by keeping only the entries i and i+1 contains empty boxes at all other positions, but a 12-SV table is defined (Definition 1.1 and §2.1) as a filling of every box of λ/μ with a nonempty subset of {1,2}. Consequently the flip and descent-resolution operations are not literally applicable to T(i) or to the modified subfilling T^{(i)} used in the definition of e_i and f_i. The paper must either extend the entire framework of Sections 2 and 3 to allow empty boxes in 12-SV tables, together with all the relevant definitions of descents, benignity, and seplist, or justify that the boxes occupied by i and i+1 always form a skew shape in the cases needed. This is a load-bearing issue for Theorems 1.2 and 1.3.","section":"§2.1 and §3.2"},{"comment":"The operator L_i is asserted to send SVRPP_n(λ) to itself and to preserve the property that the column reading word is a reverse lattice word, but no proof of either assertion is given. Since Lemma 4.5 is used to realize every partition ρ with (λ1) ⊆ ρ ⊆ ¯λ(n) as the weight of a highest weight element, the proof of Theorem 1.4 (SNP) depends on these properties. A detailed verification of the row and column inequalities under the operation L_i and of the reverse-lattice-word property of the column reading word of T* must be supplied.","section":"§4, Lemma 4.5"},{"comment":"The reconstruction algorithm in the proof of Proposition 3.12 is described informally, and its correctness is not proved. The argument should demonstrate that the procedure is deterministic and well-defined (e.g., that the 'rightmost box not filled' is always uniquely determined and that the case split on v_j > a always applies unambiguously), that it terminates, and that the resulting filling has the prescribed read(T), height(T), and ex(T). The uniqueness statement is used in Theorem 3.13 to obtain an injective embedding of crystal components into S_m^n, so the missing correctness proof is load-bearing for Theorem 1.3.","section":"§3.4, Proposition 3.12"}],"minor_comments":[{"comment":"There are several typographical errors: 'polyotpe' for 'polytope', 'coutable' for 'countable', 'crytal' and 'crytal graph' for 'crystal' and 'crystal graph', 'Grothedieck' for 'Grothendieck', 'semi-nomal' for 'seminormal', 'unmarkd content' for 'unmarked content', 'formuals' for 'formulas', 'Integeral lattice model' for 'Integrable lattice model', and 'theirin' for 'therein'.","section":"Throughout"},{"comment":"The phrase 'some (unique) box' also appears later in the paper, but uniqueness of a box containing a particular set in a column is not generally a consequence of the defining inequalities; each such use should be checked.","section":"§2.2, Definition 2.4"},{"comment":"The definition of marked multiset-valued tableaux in Definition 5.7 is clear in examples, but the marking condition 'if there exists an i located at a higher position in the same column' should be stated more precisely for boxes that contain multiple copies of i, since the row-order convention for permuting elements inside a box needs to be explicit.","section":"§5"},{"comment":"The reference list is extensive, but [15] is a self-citation that is not directly connected to the central arguments; the authors may wish to indicate its relevance in the text.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional verdict is appropriate; I identified the same soft spots in Lemma 2.9, Lemma 4.5, and Proposition 3.12. My additional concern is the empty-boxes issue for T(i) and T^{(i)}, which may require a non-trivial extension of the definitional framework in Sections 2 and 3. The claims are plausible and the paper is well written, but these gaps currently prevent the central results from being considered fully established. I do not see grounds for rejection; major revision is the appropriate outcome."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my take on arXiv:2505.19072. The paper defines a hybrid Grothendieck polynomial as a weight generating function over set-valued reverse plane partitions, with parameters t and w interpolating between the refined stable Grothendieck polynomials (set w=0) and the refined dual stable Grothendieck polynomials (set t=0). That unification is real, and the main structural results are new: symmetry via a Bender–Knuth type involution, a gl_n-crystal on SVRPPs, a Schur expansion with coefficients counted by reverse lattice words, a saturated Newton polytope theorem for straight shapes, and an omega image formula using marked multiset-valued tableaux. The paper is well organized, has worked examples, and includes a full crystal graph for a small shape in the appendix.\n\nThe proofs are mostly detailed and the constructions are explicit. The crystal operators, reading word, and the Fomin–Greene algebra argument are credible. I would not put any of the main theorems in doubt.\n\nThe soft spots are where the paper compresses routine checks in load-bearing places. The local confluence of the descent-resolution algorithm (Lemma 2.9) is the biggest one: the i = j−1 case, where the braid relation must hold for set-valued variants with {1,2} entries, is dispatched with \"it is easy to check.\" This is exactly the relation that makes the normal form unique and the involution and crystal operators well-defined. I expect the check to go through, but a referee should ask for an explicit case analysis. Similarly, Lemma 4.5 defines an operator L_i and asserts that it preserves SVRPPs and that the resulting column reading word is reverse lattice; the verification is hand-waved, and this lemma is needed for the SNP theorem. Proposition 3.12's reconstruction algorithm is convincing but stated without a formal correctness argument; it underpins the embedding of crystals into word crystals in Theorem 3.13. These are gaps in exposition, not signs of a false result. Conjecture 6.9 has a typo: both sides use ρ′ instead of ρ versus ρ′.\n\nFor people working on K-theoretic Schubert calculus, crystals, and Grothendieck polynomials, this is a useful and citable paper. It extends known techniques rather than opening a new direction, but the unification is natural and the results are precisely the ones you would want. I would send it to a serious referee, with instructions to verify Lemma 2.9 and Lemma 4.5 carefully.\n\nBest.","headline":"Solid unification of stable and dual stable Grothendieck polynomials via a new crystal on SVRPPs; main results are likely correct, but a few load-bearing local checks are asserted rather than proved.","tokens_in":34905,"tokens_out":2007,"would_cite":true,"duration_ms":19136,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces a hybrid Grothendieck polynomial over set-valued reverse plane partitions, proves its Schur expansion with reverse-lattice-word coefficients, shows straight shapes have saturated Newton polytopes, and gives a…","keywords":["hybrid Grothendieck polynomials","set-valued reverse plane partitions","crystal bases","Schur expansion","saturated Newton polytopes","omega involution","Fomin–Greene operators","reverse lattice words"],"falsifier":"Search for a benign two-letter filling of some skew shape whose columns $j-1$ and $j$ are both descents, with column $j$ mixed and containing $\\{1,2\\}$, for which resolving $j$, then $j-1$, then $j$ gives a different filling from resolving $j-1$, then $j$, then $j-1$; such a pair would falsify Lemma 2.9 and break the well-definedness of the crystal operators, and it can be checked by exhaustive search over small shapes.","tokens_in":33891,"feed_emoji":"🧮","tokens_out":6746,"duration_ms":55677,"temperature":0.7,"pith_summary":"The paper defines a two-parameter deformation of skew Schur functions, called the hybrid Grothendieck polynomial $G_{\\lambda/\\mu}(\\mathbf{x};\\mathbf{t};\\mathbf{w})$, as the weight generating function over set-valued reverse plane partitions of skew shape $\\lambda/\\mu$, with weights tracking irredundant content, column equalities, and excess. Setting all $t_i=0$ recovers the refined stable Grothendieck polynomial, and setting all $w_i=0$ recovers the refined dual stable Grothendieck polynomial, so the new object is a common parent of the two K-theoretic families. The paper proves that $G_{\\lambda/\\mu}$ is symmetric in the $\\mathbf{x}$ variables and expands into Schur functions with coefficients counted by set-valued reverse plane partitions whose reading word is a reverse lattice word. For straight shapes it shows the resulting polynomial has a saturated Newton polytope, and it gives a combinatorial formula for the image under the omega involution in the one-parameter specialization.","feed_headline":"Hybrid Grothendieck polynomials unify two K-theory families","feed_subtitle":"Set-valued reverse plane partitions give a Schur-positive crystal expansion and a saturated Newton polytope.","key_machinery":"The central objects are set-valued reverse plane partitions, fillings of a skew shape with nonempty finite sets of positive integers, weakly increasing along rows and columns, together with the three statistics ircont, ceq, and ex. The engine of the paper is an involution on two-letter set-valued reverse plane partitions obtained by flipping pure columns and then resolving descents by local rewrites on benign two-letter tables; this involution makes the crystal operators $e_i,f_i$ well-defined and independent of the order of resolution. The crystals are shown to be seminormal $\\mathfrak{gl}_n$-crystals whose highest-weight elements are exactly the set-valued reverse plane partitions whose reading word is a reverse lattice word, which yields the Schur expansion via the commuting relation $E_i(\\mathrm{read}(T))=\\mathrm{read}(e_i(T))$. For the omega image, the machinery is a family of Fomin--Greene type operators $\\tilde u_i=u_i(1+d_{\\mu,i})$ acting on partitions by adding vertical strips, whose noncommutative Cauchy identity converts the generating function into one over marked multiset-valued tableaux.","core_discovery":"For any skew shape $\\lambda/\\mu$, the hybrid Grothendieck polynomial $G_{\\lambda/\\mu}(\\mathbf{x}_n;\\mathbf{t};\\mathbf{w})$ is a Schur-positive symmetric function. Its Schur expansion is $G_{\\lambda/\\mu}=\\sum_{\\gamma,\\theta}\\mathbf{t}^{\\gamma}\\mathbf{w}^{\\theta}\\sum_{\\nu} H^{\\nu,\\gamma,\\theta}_{\\lambda/\\mu,n} s_\\nu$, where $H^{\\nu,\\gamma,\\theta}_{\\lambda/\\mu,n}$ is the number of set-valued reverse plane partitions of weight $\\nu$, column-equalities $\\gamma$, and excess $\\theta$ whose reading word is a reverse lattice word. The proof builds a $\\mathfrak{gl}_n$-crystal on set-valued reverse plane partitions whose connected components are irreducible crystals of highest weight $\\nu$, so each component contributes exactly one Schur function. As corollaries, straight-shape hybrid polynomials have saturated Newton polytopes, meaning every lattice point in the Newton polytope is an exponent vector, with degree given by the size of an explicit partition $\\bar\\lambda^{(n)}$, and the omega image of the one-parameter specialization is the generating function of marked multiset-valued tableaux.","pith_inferences":["Editorial extension: the same crystal may yield a uniform Littlewood–Richardson-type rule for the K-theoretic structure constants of Grassmannians at the hybrid level, interpolating between the known rules for the two specializations.","Editorial extension: the reverse-lattice-word criterion suggests a direct dynamic-programming algorithm for computing the coefficients $H^{\\nu,\\gamma,\\theta}_{\\lambda/\\mu,n}$ by scanning rows, making the expansion computable for shapes beyond hand-drawn examples.","Editorial extension: the unproved confluence case in Lemma 2.9 is a natural target for computer-assisted checking; a counterexample there would break the crystal operators and the Schur expansion, while the symmetric-function identities might still survive through other means.","Editorial extension: the explicit interval $(\\lambda_1)\\subseteq\\rho\\subseteq\\bar\\lambda^{(n)}$ for straight shapes suggests a conjectural analogue for skew shapes, which the paper notes is open even for the classical stable and dual stable specializations."],"forward_implications":["The hybrid polynomial is Schur positive for every skew shape, so its specializations, including the refined stable and refined dual stable Grothendieck polynomials, all have coefficients counted by reverse-lattice-word fillings.","Every straight-shape hybrid Grothendieck polynomial has a saturated Newton polytope, meaning the support of the polynomial is exactly the lattice points of its convex hull, and the Newton polytope has the integer decomposition property.","The degree of $G_\\lambda(\\mathbf{x}_n;\\mathbf{t};\\mathbf{w})$ equals the size of an explicit partition $\\bar\\lambda^{(n)}$ built from the largest strict partition inside $\\lambda$.","The omega image of $G_{\\lambda/\\mu}(\\mathbf{x};\\alpha;\\beta)$ is a generating function over marked multiset-valued tableaux, and the two classical specializations recover the known omega images of stable and dual stable Grothendieck polynomials.","Because the crystal components are irreducible $\\mathfrak{gl}_n$-crystals, the expansion coefficients are nonnegative integers and the same crystal provides a uniform combinatorial rule valid for all skew shapes at once."],"supporting_citations":[{"why":"Supplies the descent-resolution involution method that the paper extends to set-valued fillings, and defines the refined dual stable Grothendieck polynomials.","marker":"[20]"},{"why":"Defines the refined stable Grothendieck polynomials, the $t_i=0$ specialization of the hybrid polynomial.","marker":"[13]"},{"why":"Introduces set-valued tableaux and the stable Grothendieck polynomial model that set-valued reverse plane partitions generalize.","marker":"[8]"},{"why":"Provides the Littlewood–Richardson-type crystal rule for dual stable Grothendieck polynomials that the present crystal extends.","marker":"[19]"},{"why":"Introduces reverse plane partitions for dual stable Grothendieck polynomials and the weak set-valued and valued-set tableaux formulas for their omega images.","marker":"[34]"},{"why":"Fomin–Greene noncommutative Schur function theory used to derive the omega image formula.","marker":"[18]"},{"why":"Gives the saturated-Newton-polytope criterion applied in the proof of Theorem 1.4.","marker":"[48]"},{"why":"Constructs crystals on set-valued tableaux, the special case of the present crystal when $t_i=0$.","marker":"[45]"},{"why":"Earlier algebraic Schur expansion of straight-shape stable Grothendieck polynomials that the combinatorial rule extends.","marker":"[36]"}],"fun_headline_variants":["Crystal Schur expansion for hybrid Grothendieck polynomials","Hybrid Grothendieck: Schur positivity via crystal structure","Saturated Newton polytopes for hybrid Grothendieck","Hybrid Grothendieck: crystal Schur expansion and saturated polytopes","Unifying K-theory with hybrid Grothendieck crystals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire construction of the crystal operators rests on the claim that a local rewriting procedure on two-letter fillings—flipping pure columns and then resolving descents—always reaches the same final filling regardless of the order in which the rewrites are applied; the one delicate overlapping case is verified only by 'it is easy to check', so if that overlap ever failed, the operators and the Schur expansion would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Crystal Schur expansion for hybrid Grothendieck polynomials","Hybrid Grothendieck: Schur positivity via crystal structure","Saturated Newton polytopes for hybrid Grothendieck","Hybrid Grothendieck: crystal Schur expansion and saturated polytopes","Unifying K-theory with hybrid Grothendieck crystals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001457,"raw_usage":{"total_tokens":5976,"prompt_tokens":1171,"completion_tokens":4805,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":787,"completion_tokens_details":{"reasoning_tokens":4714}},"tokens_in":787,"tokens_out":4805,"duration_ms":31069,"temperature":1.0,"reasoning_tokens":4714,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:21:06.153885+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a benign two-letter filling of some skew shape whose columns $j-1$ and $j$ are both descents, with column $j$ mixed and containing $\\{1,2\\}$, for which resolving $j$, then $j-1$, then $j$ gives a different filling from resolving $j-1$, then $j$, then $j-1$; such a pair would falsify Lemma 2.9 and break the well-definedness of the crystal operators, and it can be checked by exhaustive search over small shapes.","supporting_citations":[{"cited_title":"Galashin, D","cited_arxiv_id":null,"evidence_quote":"Supplies the descent-resolution involution method that the paper extends to set-valued fillings, and defines the refined dual stable Grothendieck polynomials."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the refined stable Grothendieck polynomials, the $t_i=0$ specialization of the hybrid polynomial."},{"cited_title":"Buch, A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math","cited_arxiv_id":null,"evidence_quote":"Introduces set-valued tableaux and the stable Grothendieck polynomial model that set-valued reverse plane partitions generalize."},{"cited_title":"Galashin, A Littlewood–Richardson rule for dual stable Grothendieck poly- nomials, J","cited_arxiv_id":null,"evidence_quote":"Provides the Littlewood–Richardson-type crystal rule for dual stable Grothendieck polynomials that the present crystal extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces reverse plane partitions for dual stable Grothendieck polynomials and the weak set-valued and valued-set tableaux formulas for their omega images."},{"cited_title":"Fomin, C","cited_arxiv_id":null,"evidence_quote":"Fomin–Greene noncommutative Schur function theory used to derive the omega image formula."},{"cited_title":"Nguyen, G.N.T","cited_arxiv_id":null,"evidence_quote":"Gives the saturated-Newton-polytope criterion applied in the proof of Theorem 1.4."},{"cited_title":"Monical, O","cited_arxiv_id":null,"evidence_quote":"Constructs crystals on set-valued tableaux, the special case of the present crystal when $t_i=0$."},{"cited_title":"Lenart, Combinatorial aspects of the K-theory of Grassmannians, Ann","cited_arxiv_id":null,"evidence_quote":"Earlier algebraic Schur expansion of straight-shape stable Grothendieck polynomials that the combinatorial rule extends."}],"review_version":1}