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REVIEW 2 major objections 6 minor 21 references

Quantitative Unique Continuation on Simplex and $\mathbb{Z}^n$

T0 review · 2 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For every dimension $n\ge 3$, nonzero solutions of the discrete Schr\"odinger equation with bounded potential on $\mathbb{Z}^n$ keep values within an exponential factor of their maximum on at least $c_n L^{\lceil n/2\rceil}/\log L$ points…

desk verdict Solid high-dimensional QUC proof with a repairable operator-norm error; deserves refereeing, not desk reject. read the letter →

arxiv 2608.11602 v1 pith:OGN5MTV5 submitted 2026-08-12 math.AP math-phmath.DSmath.MP

Shihe Liu , Yunfeng Shi , Zhifei Zhang This is my paper · ORCID
classification math.APmath-phmath.DSmath.MP MSC 35B6035J1039A1205A1005E05
keywords quantitativeuniquecontinuationsimplexlatticediscreteSchr\"odingerequationPascaluncertaintyprinciplematrixshiftedSchurpolynomialsWalshdecompositioncube
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 proves a quantitative unique continuation theorem for the discrete Schr\"odinger equation on $\mathbb{Z}^n$, for every $n\ge 3$: if $u$ is a solution with bounded potential and $u(0)\neq 0$, then within the cube of side $L$ there are at least $c_n L^{\lceil n/2\rceil}/\log L$ points where $|u|$ stays above $\exp(-C_{n,K}L^{\lceil n/2\rceil+1}/\log L)|u(0)|$. It proves the analogous quantitative statement on the $n$-dimensional simplex lattice, where the number of large points is measured by the weight $\Phi_n(a)=(a_n+1)\prod_{i=1}^{n-2}(a_i+1)/(N+1)^{\lfloor(n-2)/2\rfloor}$. The engine is a quantitative Pascal uncertainty principle: a polynomial with a nonzero coefficient has many normalized coefficients, in itself and in its one-step shift, that cannot be exponentially small. A Walsh decomposition of the $\ell^1$-sphere turns the lattice problem into the simplex problem. These estimates are sharp in their cardinality up to a logarithm, and they are the stated first step toward Anderson\--Bernoulli localization in high dimensions.

What carries the argument

The load-bearing object is the quantitative Pascal uncertainty principle (Lemma 4.6 and Corollary 4.1): for a $d$-variable polynomial $p$ of total degree at most $L$ with a nonzero coefficient at exponent $\omega$, the total number of pairs $(S,\nu)$, $S$ a subset of coordinates and $\nu$ an exponent, for which the normalized coefficients of $p(x+\chi_S)$ are at least $\delta_L^d$ times the reference coefficient is at least $\prod_{i=1}^d(\omega_i+2)$. The proof constructs, from proper Pascal minors and shifted Schur polynomials, a one-variable interpolation polynomial $F$ that vanishes on the large coefficients of $p$ and $p(x+1)$; Cramer's rule and the positivity of proper Pascal minors (determinant at least $1$) control $|F|$. A Walsh decomposition of the $\ell^1$-sphere then identifies each coordinate orthant's fiber with a simplex lattice, so Theorem 2.2 transfers to $\mathbb{Z}^n$.

What would settle it

Compute the determinant of a proper Pascal minor, for example the $3\times 3$ minor with rows $r=(2,4,6)$ and columns $k=(0,2,6)$: if any proper minor has determinant $0$ or a non-integer less than $1$, the bound (4.6) in Lemma 4.3 fails and the proof from Lemma 4.5 onward no longer holds. Independently, on $\mathbb{Z}^3$ take a non-constant harmonic polynomial ($V\equiv 0$) with $u(0)\neq 0$, evaluate it on $Q_L$ for large $L$, and count the points where $|u(x)|\ge \exp(-C L^3/\log L)|u(0)|$; a count below a positive constant times $L^2/\log L$ would refute Theorem 2.4's conclusion.

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Extended reading notes

Core claim

The central claim is that quantitative unique continuation holds, with explicit rates, on both the simplex lattice $\Delta_N^{(n)}$ and the lattice cube $Q_L$ for arbitrary $n\ge 3$. Specifically, Theorem 2.2 states that if $g:\Delta_N^{(n)}\to\mathbb{R}$ satisfies $g(a)\neq 0$ and $|\sum_{i=1}^n g(\beta+e_i)|\le e^{-C_n N}|g(a)|$ for every $\beta\in\Delta_{N-1}^{(n)}$, then at least $c_n\Phi_n(a)$ points $\alpha$ satisfy $|g(\alpha)|\ge e^{-C_n N}|g(a)|$, with $\Phi_n(a)$ as above. Theorem 2.4 states that if $\Delta_{\mathbb{Z}^n}u=Vu$ on $Q_L$ with $\|V\|_{\ell^\infty}\le K$ and $u(0)\neq 0$, then at least $c_n L^{\lceil n/2\rceil}/\log(2+L)$ points $x\in Q_L$ satisfy $|u(x)|\ge \exp(-C_{n,K}L^{\lceil n/2\rceil+1}/\log(2+L))|u(0)|$. The simplex decay is exponential in $N$ with no logarithmic loss; the lattice result carries an extra $\log(2+L)$ factor in both the decay and the cardinality.

Load-bearing premise

The whole argument hinges on the combinatorial fact, cited from [Li26a], that every proper minor of the Pascal matrix has determinant at least 1; if a single proper minor had determinant 0 or a non-integer value below 1, the interpolation polynomial used to control the exponential bounds would not exist with the required size, and the chain from the Pascal principle to the lattice theorem would break.

Editorial extensions

If this is right

  • Taking $a=R\mathbf{1}_n$ and $N=nR$ in the simplex theorem gives Theorem 2.1: a function on $\Delta_{nR}^{(n)}$ satisfying the centered decay condition must be $\ge e^{-C_n R}|g(R\mathbf{1}_n)|$ on at least $c_n R^{\lceil n/2\rceil}$ points.
  • Theorem 2.4 answers the support-cardinality question for $\mathbb{Z}^n$ up to a $(\log L)^{-1}$ factor: the lower bound $L^{\lceil n/2\rceil}$ is optimal in power of $L$, as the paper notes via the cited sharpness proposition.
  • All the main estimates hold for complex-valued $u$ and $V$ with unchanged constants, since the proof runs line by line in the complex case (Remark 2.2).
  • The exponents $a=\lceil n/2\rceil+1$ and $b=\lceil n/2\rceil$ lie outside the range needed for Anderson\--Bernoulli localization by the standard multiscale argument, so the paper does not claim localization; it supplies the quantitative unique continuation ingredient in that range.

Reading between the lines

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

  • Editorial inference: the quantitative Pascal uncertainty principle is a self-contained statement about polynomials, independent of Schr\"odinger equations; it should apply to any constant-coefficient difference equation whose symbol gives a Pascal-type shift, so similar QUC theorems are plausible for other finite-range hopping operators.
  • Editorial inference: the simplex theorem's exponential-in-$N$ decay with no logarithmic loss suggests that the $(\log L)^{-1}$ factor in the $\mathbb{Z}^n$ theorem comes from the Walsh/Voronoi layer, not from the Pascal principle; sharpening the geometric pigeonholing would remove or reduce the log.
  • Editorial inference: a natural stress test is numerical: on $\mathbb{Z}^3$ with $V=0$, count the large-value set for a generic harmonic polynomial; the proof's constants are not tracked, so the test would reveal whether the predicted power $L^2/\log L$ appears with moderate constants or only asymptotically.
  • Editorial inference: the weight $\Phi_n(a)$ singles out the smallest coordinate $a_n$; this predicts that solutions whose large initial value sits near the boundary of the simplex (small $a_n$) have fewer guaranteed large points, a direction that could be tested by constructing explicit $g$ from binomial-coefficient polynomials.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper proves quantitative unique continuation (QUC) on the n-dimensional simplex lattice Δ_N^{(n)} and on Z^n for every n≥3. Theorem 2.2 gives a lower bound c_n Φ_n(a) on the number of points at which a function satisfying the approximate recurrence (2.4) is not too small; Theorem 2.4 transfers this to the Schrödinger equation Δu=Vu in a cube, with decay exponent ⌈n/2⌉+1 and cardinality exponent ⌈n/2⌉ up to a logarithmic factor. The proof chain is: (i) a polynomial dictionary identifying the exact recurrence with DF=0; (ii) a quantitative Pascal uncertainty principle built from shifted Schur polynomial interpolation and proper Pascal minors; (iii) shell estimates on the simplex; (iv) a stability argument using the projection T_{A_1} onto ker D; and (v) a Walsh decomposition of ℓ1-spheres together with a Voronoi double-counting argument for Z^n. The overall architecture is coherent and the chain is checkable, but one load-bearing operator-norm computation in Step (iv) is incorrect as stated.

Significance. If the corrected operator-norm estimate is substituted, the paper's main theorems appear to follow, giving the first QUC on the simplex and on Z^n for arbitrary n≥3, extending [LZ22] and answering [Li26b, Conjecture 1.2] up to a (log L)^{-1} factor. The quantitative Pascal uncertainty principle and the Walsh/Voronoi reduction are useful new tools. The paper is explicit about the exponents and honestly notes that they fall short of the Anderson-Bernoulli localization threshold. The proof is essentially self-contained, apart from an imported determinant-positivity fact from [Li26a] that is used in a load-bearing way.

major comments (2)
  1. [§6.2, Theorem 6.4, Eqs. (6.31) and (6.36)] The claimed operator norm equality is false. With a=(n-2)/n, the integral in (6.31) evaluates to N/n ∫_0^1 (1+at)^{N-1} dt = [(1+a)^N -1]/ (n-2) = [(2-2/n)^N -1]/(n-2), not (2-2/n)^{N-1}/(n-2). The lower-bound computation in (6.36) gives the same corrected value, so the displayed K_{n,N} in (6.27) is not the operator norm. Consequently the inequality (6.40), which uses this K_{n,N}, is not justified as written, and the passage from the exact-recurrence function h to the original function g in (6.41)-(6.43) has a genuine gap. The error is repairable: replacing K_{n,N} by [(2-2/n)^N -1]/(n-2) gives the same exp(O_n(N)) behavior, and enlarging C_n in (6.40) restores the proof. The manuscript must correct this computation and all statements that depend on it.
  2. [§4.1, Lemma 4.3 and Eq. (4.7)] The proof relies on the strict positivity of proper Pascal minors, imported from [Li26a, Lemma 4.1] without statement or proof. This fact is load-bearing: it gives det P ≥ 1, which is used to define the interpolating polynomial via Cramer's rule and to control |F(t)| in (4.6), and the bounds then propagate to Theorem 4.4, Lemma 4.5, Lemma 5.1, and the main theorems. The manuscript should either prove this determinant positivity in an appendix or state it explicitly as an imported lemma, rather than referring to it inside a proof with no statement of the result.
minor comments (6)
  1. [§5, first paragraph] The sentence 'by Lemma 6.3, (2.8) is equivalent to DF=0' should refer to Lemma 3.1, which is the lemma that establishes this equivalence.
  2. [§6.3, proof of Theorem 2.2] The line 'By Lemma , assumption (2.4) is exactly...' has a missing lemma reference; it should cite Lemma 3.1.
  3. [§2.1, Remark 2.1] In the n=2 remark, the range '0≤j≤2R' is undefined; it should be '0≤j≤N'.
  4. [§7, first paragraph] The text says 'QUC for stationary Schrödinger equation on Z^d'; the symbol should be Z^n for consistency with the rest of the paper.
  5. [§4.1, proof of Lemma 4.3] In the definition of the reversal matrix J_q, the range '1≤i,j≤1' should be '1≤i,j≤q'.
  6. [Throughout] The paper contains numerous typos, including 'equavalent', 'fucntion', 'knowlegdes', 'pespective', 'aslo', and 'descr'; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the core results are derived in-paper from first-principles lemmas; the cited external facts are independent combinatorial inputs, and the self-citations are not load-bearing.

full rationale

The paper's central chain is Theorem 2.4 -> Theorem 2.2 via the stability argument -> Theorem 2.3 via the quantitative Pascal uncertainty principle -> Lemmas 5.1, 4.6, 4.5, Theorem 4.4, and Lemma 4.3. Each step is proved inside the paper: the ker D projection (Lemma 6.3), the Walsh transform (Theorem 7.1), the shell estimates (Lemma 5.1), and the quantitative Pascal UP (Lemmas 4.5 and 4.6) are all constructed and justified with displayed equations. The only external inputs are (i) the LGV determinant theorem, stated and sketched in Theorem 3.4; (ii) the proper-Pascal-minor positivity cited from [Li26a, Lemma 4.1] and used in Lemma 4.3; and (iii) the shifted-Schur identity, which the paper derives from its own Lemma 4.2. None of these is equivalent to the target QUC statements, and none is fitted to the conclusions. The self-citations [LSZ26a] and [LSZ26b] appear only in the introduction and in Remark 5.1 as interpretive remarks; the actual estimates around (5.3)-(5.13) are derived in the paper from the multinomial theorem and the quantitative Pascal UP without relying on the cited work. The citation to [Li26a] is not a self-citation, since its author is not among Liu, Shi, and Zhang. Finally, the skeptical observation about Theorem 6.4 concerns a numerical slip in the claimed equality of the operator norm; even if the exact constant were wrong, the displayed quantity remains an upper bound of the same exponential type, so the stability argument is not circular. There is no fitted parameter renamed as a prediction and no uniqueness theorem imported from the authors' prior work; the score is therefore 1, reflecting only a minor non-load-bearing self-citation in expository remarks.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The theorems are existence results for constants c_n,C_n, so the ledger contains no fitted numerical parameters. The mathematical machinery (Walsh basis, kernel projection C_{n,N}) is constructed and verified inside the paper, hence recorded as invented entities with internal evidence. The axioms are standard combinatorial tools (LGV, shifted Schur identity) plus the cited proper-Pascal-minor positivity, together with the domain assumptions defining the discrete Laplacian and the polynomial dictionary.

assumptions (5)
  • standard math Lindström-Gessel-Viennot determinant theorem (path cancellation)
    Used in Lemma 4.3 to justify positivity of proper Pascal minors and in Appendix A to prove the factorial bialternant identity (Lemma 4.2).
  • standard math Shifted Schur function tableau identity (4.2) of Okounkov-Olshanski
    Imported from [OO97, Theorem 11.1] to prove Lemma 4.1's monotonicity of shifted Schur polynomials.
  • standard math Proper Pascal minors have determinant at least 1 (Li26a, Lemma 4.1)
    Load-bearing in Lemma 4.3: guarantees the interpolation polynomial F exists and |F(t)| obeys the product bound; the proof is only cited, not reproduced.
  • domain assumption Discrete Laplacian convention (2.10) and the nearest-neighbour recurrence (2.8) model the problem
    Theorems 2.2-2.4 are stated for these equations; all subsequent polynomial and Walsh arguments use these normalizations.
  • domain assumption Factorial-normalized polynomial dictionary (Lemma 3.1) faithfully represents simplex functions
    The mapping g ↦ F(z)=∑ g(α)z^α/α! with D=∂_{z_1}+...+∂_{z_n} identifies the recurrence with DF=0; the entire proof is conducted in this dictionary.
invented entities (2)
  • Walsh basis v_{J,b} on the ℓ1-sphere with lowering operator D independent evidence
    purpose: Decomposes functions on the sphere S_r into 2^n simplex sectors, allowing QUC on Z^n to be derived from QUC on Δ^{(n)}_r via the transform identities (7.9)-(7.10).
    The basis and its properties are explicitly constructed and proved in Section 7.1; no external evidence is needed. It is a proof device, not a physical postulate.
  • Projection operator C_{n,N} and the A_t homotopy decomposition (Lemma 6.3) independent evidence
    purpose: Splits any homogeneous polynomial F into an exactly-recurrent part H=F(A_1 z) in ker D plus a remainder S(z)Q(z), and bounds the remainder norm exactly via Theorem 6.4.
    Fully constructed and verified in Section 6.2; its operator norm is computed in Theorem 6.4 with an extremal example, so it is a self-contained mathematical tool.

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Cite this review

Pith. "Pith review of Quantitative Unique Continuation on Simplex and $\mathbb{Z}^n$." pith.science (2026). https://pith.science/paper/OGN5MTV5

@misc{pith2026260811602,
  author       = {Pith},
  title        = {Pith review of: Quantitative Unique Continuation on Simplex and $\mathbbZ^n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGN5MTV5}},
  note         = {Machine review of arXiv:2608.11602}
}
abstract

In this work, we establish the quantitative unique continuation on both $n$-dimensional simplex lattice $\Delta^{(n)}_{N}$ and $\mathbb{Z}^n$ for arbitrary $n\geq 3$.

Figures

Figures reproduced from arXiv: 2608.11602 by the authors.

Figure 1
Figure 1. The corresponding Young diagram and a semistandard Young tableau of partition λ = (4, 3, 3, 1). 4.1. Shifted Schur polynomials and several polynomial interpolation results. We first introduce some combinatorial results which will be useful in our proof. Let λ be a partition of length at most q. Define the Shifted Schur polynomial s ∗ λ (x1, . . . , xq) as s ∗ λ (x1, . . . , xq) := det[(xi + q − i)λj+q−j ]1≤i,j≤q det… view at source ↗
Figure 2
Figure 2. Geometric interpretation of decomposition F = TA1 F + (I − TA1 )F and Theorem 6.4. For E ∈ H(n) N−1 , modeling after (6.15) and (6.16), we define the operator Cn,N : H (n) N−1 → H(n) N by (Cn,N E)(z) := Z 1 0 ℓ(z)E(Atz) dt = Z ℓ(z) 0 E [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. A two-dimensional schematic of (7.44) and Lemma 7.5. Lemma 7.5 (Voronoi estimate). Let S ⊂ QL/2 contain the origin, and set ZL = Q⌊L/(8n)⌋ . Suppose that for every z ∈ ZL and every x ∈ NS(z), we have mS(z) ≳n ϕn(|x1 − z1|, . . . , |xd − zd|). (7.45) Then |S| ≳n L q log(2 + L) . (7.46) Here q = ⌈n/2⌉. Proof. For fixed z, (7.45) gives X x∈NS(z) 1 ϕn(|x1 − z1|, . . . , |xd − zd|) ≳n mS(z) mS(z) = 1. Sum over z ∈ ZL and… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: A schematic picture of the directed graph G for q = 3 and λ = (3, 2, 1). Circles denote the starting points Pi , squares denote the terminations Qi , and the colored edges show one identity-matched vertex-disjoint path family. Set Pi := (i, q − i + 1), Qj := (q, q − j …

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Works this paper leans on

21 extracted references · 13 canonical work pages

  1. [1]

    Voronoi diagrams---a survey of a fundamental geometric data structure

    Franz Aurenhammer. Voronoi diagrams---a survey of a fundamental geometric data structure. ACM Computing Surveys , 23(3):345--405, 1991

    work page 1991

  2. [2]

    Bourgain and C

    J. Bourgain and C. E. Kenig. On localization in the continuous A nderson- B ernoulli model in higher dimension. Invent. Math. , 161(2):389--426, 2005

    2005

  3. [3]

    Buhovsky, A

    L. Buhovsky, A. Logunov, E. Malinnikova, and M. Sodin. A discrete harmonic function bounded on a large portion of Z^2 is constant. Duke Math. J. , 171(6):1349--1378, 2022

    2022

  4. [4]

    Bou-Rabee, W

    A. Bou-Rabee, W. Cooperman, and S. Ganguly. Unique continuation on planar graphs. Discrete Anal. , pages Paper No. 16, 12, 2025

    work page 2025

  5. [5]

    Ding and C

    J. Ding and C. K. Smart. Localization near the edge for the A nderson B ernoulli model on the two dimensional lattice. Invent. Math. , 219(2):467--506, 2020

    2020

  6. [6]

    The graph voronoi diagram with applications

    Martin Erwig. The graph voronoi diagram with applications. Networks , 36(3):156--163, 2000

    work page 2000

  7. [7]

    Counterexamples to the landis conjecture in dimensions three and higher

    Rupert L Frank and Paata Ivanisvili. Counterexamples to the landis conjecture in dimensions three and higher. arXiv preprint arXiv:2608.00802 , 2026

    arXiv 2026

  8. [8]

    Binomial determinants, paths, and hook length formulae

    Ira Gessel and G\' e rard Viennot. Binomial determinants, paths, and hook length formulae. Adv. in Math. , 58(3):300--321, 1985

    work page 1985

Show all 21 references
  1. [9]

    Jitomirskaya

    S. Jitomirskaya. Ergodic S chr\" o dinger operators (on one foot). In Spectral theory and mathematical physics: a F estschrift in honor of B arry S imon's 60th birthday , volume 76 of Proc. Sympos. Pure Math. , pages 613--647. Amer. Math. Soc., Providence, RI, 2007

    2007

  2. [10]

    On the lowest possible dimension of supports of solutions to the discrete schrodinger equation

    Stanislav Krymskii. On the lowest possible dimension of supports of solutions to the discrete schrodinger equation. arXiv preprint arXiv:2401.02800 , 2024

    2024 arXiv

  3. [11]

    L. Li. Anderson- B ernoulli localization at large disorder on the 2 D lattice. Comm. Math. Phys. , 393(1):151--214, 2022

    2022

  4. [12]

    Discrete unique continuation on simplex

    Linjun Li. Discrete unique continuation on simplex. arXiv preprint arXiv:2608.02707 , 2026

    2026 arXiv

  5. [13]

    On support cardinality for the discrete schr \"o dinger equation: L

    Linjun Li. On support cardinality for the discrete schr \"o dinger equation: L. li. Letters in Mathematical Physics , 116(4):90, 2026

    2026

  6. [14]

    On the vector representations of induced matroids

    Bernt Lindstr\" o m. On the vector representations of induced matroids. Bull. London Math. Soc. , 5:85--90, 1973

    1973

  7. [15]

    Logunov, E

    A. Logunov, E. Malinnikova, N. Nadirashvili, and F. Nazarov. The L andis conjecture on exponential decay. Invent. Math. , 241(2):465--508, 2025

    2025

  8. [16]

    S. Liu, Y. Shi, and Z. Zhang. Anderson localization for the hierarchical A nderson- B ernoulli model on Z ^d . arXiv preprint arXiv:2604.18989 , 2026

    2026 arXiv

  9. [17]

    Localization and unique continuation for the anderson-bernoulli model with long-range hopping on Z

    Shihe Liu, Yunfeng Shi, and Zhifei Zhang. Localization and unique continuation for the anderson-bernoulli model with long-range hopping on Z . arXiv preprint arXiv:2607.03472 , 2026

    2026 arXiv

  10. [18]

    Der-Tsai Lee and C. K. Wong. Voronoi diagrams in L_1 ( L_ ) metrics with two-dimensional storage applications. SIAM Journal on Computing , 9(1):200--211, 1980

    1980

  11. [19]

    Li and L

    L. Li and L. Zhang. Anderson- B ernoulli localization on the three-dimensional lattice and discrete unique continuation principle. Duke Math. J. , 171(2):327--415, 2022

    2022

  12. [20]

    Spatial Tessellations: Concepts and Applications of Voronoi Diagrams

    Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, and Sung Nok Chiu. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams . John Wiley & Sons, Chichester, 2 edition, 2000

    2000

  13. [21]

    Shifted schur functions

    Andrei Okounkov and Grigori Olshanski. Shifted schur functions. Algebra i analiz , 9(2):73--146, 1997

    1997

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