Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta values

Henrik Bachmann; Steven Charlton

REVIEW 2 major objections 5 minor 10 references

Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta values

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves a generalized Jacobi-Trudi determinant identity for regularized Schur multiple zeta values, and derives that every checkerboard-style value with alternating entries 1 and 3 is a polynomial in Riemann zeta values.

desk verdict Solid determinant generalization; checkerboard purity claims rest on an unproved subribbon-closure assertion that needs a fix. read the letter →

arxiv 1908.05061 v1 pith:6Y2AZ6WV submitted 2019-08-14 math.NT math.CO

Henrik Bachmann , Steven Charlton This is my paper · ORCID

classification math.NTmath.CO MSC 11M3205E05

keywords SchurmultiplezetavaluesJacobi-TrudideterminantsregularizationcheckerboardtableauxribbondecompositionsoutsideRiemann

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

This paper establishes a determinant identity for regularized Schur multiple zeta values: for a skew Young diagram with constant diagonal entries, the value equals a determinant of regularized values attached to sub-ribbons of a fixed ribbon (Theorem 3.3). This generalizes the classical Jacobi-Trudi formulae, which are recovered when the fixed ribbon is a row or a column. The identity is then used to prove that every checkerboard-style Schur multiple zeta value with alternating entries 1 and 3 is a polynomial in $\pi^4$ and odd Riemann zeta values (Theorem 4.3). The same determinant yields shape conditions under which such values are pure polynomials in $\pi^4$ only, or in odd zeta values only, and analogous conditions for alternating entries 1 and 2 (Corollaries 4.5 and 4.8). A reader should care because the paper turns an infinite family of complicated nested sums into a determinant whose entries are known constants.

What carries the argument

The load-bearing mechanism is an outside decomposition of a skew diagram into ribbons: a way of cutting the diagram into ribbon-shaped pieces, each starting on the left or bottom border and ending on the right or top border. From such a decomposition one forms a containing ribbon $R_\Theta$ and its sub-ribbons $R_\Theta(i,j)$ defined by intervals of diagonal content; the generalized sum $S_M^f(k)$ over semi-standard tableaux with weight $f(m,d)$ specializes to truncated Schur multiple zeta values via $f(m,d)=m^{-d}$. Theorem 2.8 proves the determinant identity for $S_M^f$ by a lattice-path argument whose step weights depend on horizontal position, and Lemma 3.1 transfers the identity to the regularized setting, producing Theorem 3.3. Every later checkerboard result is a deduction from this single determinant identity.

What would settle it

Construct the outside decomposition of Corollary 2.10 for an admissible 1-3 checkerboard tableau satisfying a pure-tessellation condition, with the stair as the chosen ribbon, and inspect every sub-ribbon: if any sub-ribbon is not of the same stair type, the purity claim fails. A high-precision numerical evaluation of such a tableau that lands outside the claimed polynomial ring would also refute the result.

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Extended reading notes

Core claim

The paper's central claim is that regularized Schur multiple zeta values satisfy a generalized Jacobi-Trudi determinant formula (Theorem 3.3): $$\zeta_{\mathrm{reg}}(k) = \det\!\left(\zeta_{\mathrm{reg}}(R_{\Theta}^{k}(i,j))\right)_{1\le i,j\le n}$$ for an edge-connected skew diagram $\lambda/\mu$ with outside decomposition $\Theta=(\theta_1,\dots,\theta_n)$ and a Young tableau $k$ with constant diagonal entries; the matrix entry is $0$ when the sub-ribbon $R_{\Theta}(i,j)$ is undefined. Taking the fixed ribbon to be a row or a column recovers the earlier Jacobi-Trudi formulae for Schur multiple zeta values. Applied to checkerboard-style tableaux with alternating 1 and 3 entries, the identity proves that every such regularized value lies in $\mathbb{Q}[\pi^4,\zeta(3),\zeta(5),\dots][T]$ (Theorem 4.3). It further shows that when the shape is tessellated purely by one stair type, the value is pure: $\mathbb{Q}[\pi^4]$ for S- and $S^\star$-type stairs, $\mathbb{Q}[\zeta(4n+1)\mid n\ge1]$ for A-type stairs, and $\mathbb{Q}[\zeta(4n+3)\mid n\ge0]$ for B-type stairs (Corollary 4.5), with parallel purity conditions for alternating 1 and 2 entries (Corollary 4.8).

Load-bearing premise

The proof assumes, without a separate argument, that when a shape is tiled by one kind of stair, the determinant construction produces only sub-stairs of that same kind; if any other kind appeared, the determinant would introduce extra zeta values and the clean purity conclusions would fail.

Editorial extensions

If this is right

Every checkerboard-style Schur multiple zeta value with alternating entries 1 and 3 can be written as a polynomial in $\pi^4$ and odd Riemann zeta values, so this whole family of infinite sums is reducible to classical constants.
Any such tableau tessellated purely by S- or $S^\star$-type stairs evaluates into $\mathbb{Q}[\pi^4]$; for A-type stairs it evaluates into $\mathbb{Q}[\zeta(4n+1)\mid n\ge1]$, and for B-type stairs into $\mathbb{Q}[\zeta(4n+3)\mid n\ge0]$.
The same determinant mechanism yields purity conditions for alternating entries 1 and 2: S-type stairs give $\mathbb{Q}[\zeta(3n)\mid n\ge1]$, $S^\star$-type stairs give $\mathbb{Q}[\zeta(3n)\mid n\text{ odd}]$, and A-type stairs give $\mathbb{Q}[\zeta(3n+1)\mid n\ge1]$.
The explicit determinant evaluation of a 3-by-3 square in the paper shows that the formula is a practical route from a shape to a closed polynomial expression.
The observed product-minus-gluing phenomenon in checkerboard work becomes a direct determinant consequence: $B_{1,3}(n-1)A_{1,3}(n)-G_{1,3}(n)$ is a rational multiple of $\pi^{8n}$ with an explicit coefficient.

Reading between the lines

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

Editorial extension: the same content-dependent lattice-path argument should yield determinant identities for any family of tableaux sums whose weights factor by diagonal content, potentially giving evaluations for entry patterns beyond the one-three and one-two checkerboards.
Editorial extension: the purity pattern suggests the general principle that if all building-block stair values of a tiling lie in some subring, then every shape tessellated by those stairs lies in that subring; the testable combinatorial core is the claim that all sub-ribbons in the outside decomposition inherit the stair type.
Editorial extension: the interpolation remark in the paper points to a concrete experiment—add a $t$-deformation to the weights and check whether the determinant identity survives; if it does, the checkerboard evaluations should interpolate between the regularized and shuffle-regularized families.

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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 generalized Jacobi-Trudi determinant (Theorem 3.3) is the real contribution, and it's solid. The paper adapts Hamel-Goulden's outside-decomposition framework to the functions S^f_M, with edge weights that depend on content, then passes to regularized Schur multiple zeta values via the IKZ asymptotics. This genuinely extends the row and column formulae in NPY, and the proof is a standard Gessel-Viennot variation. I didn't find circularity here. Corollary 3.4, which says any such value can be written as a polynomial in subribbon values for any chosen ribbon R, is a nice and useful tool.

The application to 1-3 checkerboards is mostly there. Theorem 4.3, that all regularized 1-3 checkerboards lie in Q[pi^4, zeta(3), zeta(5), ...][T], follows by choosing a long column and using the prior BY evaluations. That logic is sound and not circular. Proposition 4.10's gluing formula is also a natural consequence of the determinant, and Lemma 4.9's Bernoulli evaluation can be checked directly.

Now the soft spot, and I think the stress-test concern lands. Proposition 4.4 asserts that if a checkerboard shape is tessellated purely by F-type stairs and you choose R to be that same F-type stair, then every subribbon R_Theta(i,j) from the Corollary 2.10 outside decomposition is also an F-type stair. The proof says 'the construction ... assures' but gives no argument. This is load-bearing: Corollaries 4.5 and 4.8 claim odd/even purity precisely because every determinant entry is F_{1,3}(n) or F_{1,2}(n). Proposition 4.10 is a useful counterpoint: with an A-type ribbon, the 2x2 determinant has off-diagonal entries S and S*, so subribbon type is not automatic from the ribbon being of one type. That doesn't refute the pure-F case, but it shows the claim needs proof rather than assertion. If even one subribbon came out as a different stair type, the determinant would contain other zeta values and the purity conclusion would fail.

This is a genuine gap, but it is fixable. The rest of the paper holds together. Minor issues: Lemma 4.7's telescoping is terse, and the proof of Theorem 2.8 is compressed, but neither is damaging.

Who is this for? People working on Schur multiple zeta values and symmetric-function determinants. The determinant formula deserves to be cited on its own. I would send this to a serious referee, with the instruction to push for a proof or a repaired statement of Proposition 4.4. A moderately revised version should be publishable.

Referee Report

2 major / 5 minor

Summary. The paper proves a generalized Jacobi-Trudi determinant formula for regularized Schur multiple zeta values. Theorem 2.8 establishes a determinant identity for a general weighted sum S^f_M over semi-standard Young tableaux with an outside decomposition, via a Gessel-Viennot path argument in the style of Hamel-Goulden; Theorem 3.3 specializes this to regularized Schur multiple zeta values zeta_reg. The applications target checkerboard-style tableaux with alternating entries: Theorem 4.3 shows that every 1-3 checkerboard Schur multiple zeta value lies in Q[pi^4, zeta(3), zeta(5), ...][T], Proposition 4.4 and Corollaries 4.5 and 4.8 give purity conditions, and Proposition 4.10 evaluates a gluing product. The determinant identity is the central tool; the purity results depend on Corollary 3.4 and on the claim in Proposition 4.4 that subribbons of an F-type stair are again F-type.

Significance. The determinant theorem is a genuine and useful extension of the Jacobi-Trudi formulae in [NPY], and the regularization argument via [IKZ] is standard and sound. The paper also gives explicit evaluations, including Bernoulli-polynomial expressions, and answers questions raised in [BY]. The Gessel-Viennot proof of Theorem 2.8 is plausible, and Theorem 3.3 is well supported. However, the abstract's claims about purely odd or purely even zeta values rest on Proposition 4.4, whose key closure assertion is unproved. The significance is therefore conditional; with a proof of that combinatorial assertion, the paper would be a solid contribution to the subject.

major comments (2)

[§4, Proposition 4.4] Proposition 4.4 is the load-bearing step for the purity statements in Corollaries 4.5 and 4.8, but its proof contains the unsupported sentence: 'The construction of the outside decomposition Theta in the proof of Corollary 2.10 then assures that all the subribbons R_Theta(i,j) are also F-type stairs.' No argument is given that the cut points produced by the construction preserve the stair type. This is not a formal consequence of the determinant theorem; Proposition 4.10 shows that subribbons of a ribbon of one type can be of other types when the outside decomposition contains a different stair type. A proof is needed that a pure F-tessellation forces all theta_i and hence every R_Theta(i,j) to be F-type. If this fails, the determinant in Corollary 3.4 would contain entries outside Q[F_{a,b}(n)] and the conclusions of Corollaries 4.5 and 4.8 would be false.
[§4, Lemma 4.7] The final step of Lemma 4.7 says only that the desired identity 'follows directly by the harmonic product formula ... since terms in the second summation telescope.' Since Lemma 4.7 is used to prove Corollary 4.8(iii), the telescoping should be displayed or replaced by a short induction; as written, the proof is not checkable from the text.

minor comments (5)

[§4, Proposition 4.4 and Corollaries 4.5/4.8] The phrase 'tessellated purely by F-type stairs' is used without a formal definition; a precise definition, for example in terms of an outside decomposition all of whose parts are translates of a fixed stair shape, would make the statement and the proof checkable.
[§4, Proposition 4.2] The symbol zeta*({1,3}^n) is not defined; if it denotes the zeta-star value or a star-regularized value, please clarify.
[§2, proof of Theorem 2.8] The path construction is described briefly; adding a precise statement of the bijection between path systems and semi-standard Young tableaux, or a small weighted example, would help the reader verify the modified edge weight f(j, i-1).
[§4, definition of G_{1,3}(n)] The displayed definition of G_{1,3}(n) contains stray TeX artifacts such as 'bracehtipdownleft' and 'bracehtipupright'; the formula should be typeset cleanly.
[§4, Lemma 4.9] The proof states 'One can check directly' for the generating-function identity involving Bernoulli polynomials; a few intermediate manipulations would improve verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Jacobi-Trudi determinant is derived from Gessel-Viennot and does not assume the checkerboard conclusions; reliance on [BY] is a prior independent evaluation. The main caveat is an unproved combinatorial closure assertion in Proposition 4.4, which is a correctness gap rather than a circular reduction.

full rationale

The central identity Theorem 3.3 is obtained from Theorem 2.8, which is proved by adapting the Hamel-Goulden/Gessel-Viennot lattice-path argument: edge weights are changed from x_j to f(j, i-1), and then Lemma 3.1 transfers the identity from truncated sums to regularized polynomials. This derivation does not use the checkerboard evaluations it later feeds. Theorem 4.3 reduces arbitrary 1-3 checkerboard values to the four column-type entries ζ({1,3}^n), ζ(3,{1,3}^n), ζreg({3,1}^n), and ζreg(1,{3,1}^n), and those base values are supplied by (4.1), (4.2), and Proposition 4.2. The underlying evaluations in [BY] are prior, explicitly stated, parameter-free results with their own proofs; the fact that the first author is also an author of [BY] is self-citation, but under the review rules it is independent support because it does not assume the target result and can be checked externally. Proposition 4.4 is the one weak point: it asserts, without proof, that the outside decomposition of a pure F-tessellated diagram has all subribbons F-type ('The construction of the outside decomposition Θ in the proof of Corollary 2.10 then assures that all the subribbons RΘ(i,j) are also F-type stairs'). Proposition 4.10 shows that subribbons of an A-type ribbon can be S- and S*-type, so the closure property is not automatic from the definition of subribbon. This missing combinatorial proof affects the purity conclusions in Corollaries 4.5 and 4.8, but it is an unproved geometric claim, not a reduction of the conclusion to its own input; hence it does not make the paper circular. No fitted parameter is relabeled as a prediction, and no uniqueness theorem from the authors is imported to force an ansatz. The determinant machinery is self-contained against the Gessel-Viennot framework, so the paper receives score 0 on circularity, with the noted non-circular correctness caveat.

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

The central determinant theorem is self-contained apart from the standard Gessel-Viennot path machinery imported from [HG]; the applications additionally draw on evaluations from [BY] and classical multiple zeta facts. No free parameters or invented entities appear.

assumptions (5)

standard math Hamel-Goulden outside decomposition theorem for Schur functions (Theorem 3.1 in [HG])
The proof of Theorem 2.8 is a variation of this result and imports its Gessel-Viennot path machinery.
standard math Ihara-Kaneko-Zagier asymptotic expansion of truncated multiple zeta values as polynomials in log(M)+gamma plus O(log^J M / M)
Used in Lemma 3.1 to define the regularized SZV and in Theorem 3.3 to pass from finite-M determinant identities to zeta_reg identities.
domain assumption The semi-standard Young tableau inequalities and the harmonic product for Schur multiple zeta values
Used to decompose zeta_M(k) into linear combinations of truncated MZVs and to apply [IKZ] and [BY] evaluations.
domain assumption Evaluations of checkerboard stairs S1,3, S*1,3, A1,3, B1,3 and the formula zeta(3,{1,3}^n) from [BY]
These prior results by Bachmann and Yamasaki are used as inputs in Proposition 4.2 and Theorem 4.3.
standard math Duality and harmonic product formulas for multiple zeta values, plus the 1-3 formula zeta({1,3}^n)=2 pi^{4n}/(4n+2)!
Used in Proposition 4.2, Lemma 4.7, and Corollary 4.8.

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Pith. "Pith review of Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta values." pith.science (2026). https://pith.science/paper/6Y2AZ6WV

@misc{pith2026190805061,
  author       = {Pith},
  title        = {Pith review of: Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6Y2AZ6WV}},
  note         = {Machine review of arXiv:1908.05061}
}

read the original abstract

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.

Figures

Figures reproduced from arXiv: 1908.05061 by the authors.

**Figure 2.** Starting with a skew diagram λ/µ = (4, 3, 3, 2, 1)/(1) and a ribbon R = (6, 6, 6, 5, 3)/(6, 6, 4, 2), we can cover the plane by R and consider its intersections with λ/µ. This gives an outside decomposition Θ = (θ1, θ2, θ3, θ4) with RΘ = R. 3. Jacobi-Trudi formula for regularized Schur multiple zeta values We now recall the definition of Schur multiple zeta values, which were first defined in [NPY]. For a Young tabl… view at source ↗

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Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Bachmann: Interpolated Schur multiple zeta values , J

H. Bachmann: Interpolated Schur multiple zeta values , J. Aust. Math. Soc., 104 , 2018, 289--307

work page 2018
[2]

Bachmann, Y

H. Bachmann, Y. Yamasaki: Checkerboard style Schur multiple zeta values and odd single zeta values , Math. Zeitschrift, 290 (3), 2018, 1173--1197

work page 2018
[3]

Gessel, G

I. Gessel, G. X. Viennot: Determinants, paths, and plane partitions , 1989, preprint

work page 1989
[4]

G. Z. Giambelli: Alcune proprieta dele funzioni simmetriche caratteristiche , Atti Torino 38 , 1903,823--844

work page 1903
[5]

Hamel, I

A.M. Hamel, I. Goulden: Planar decompositions of tableaux and Schur function determinants , European J. Combin. 16 no. 5, 1995, 461--477

work page 1995
[6]

Hoffman, K

M. Hoffman, K. Ihara: Quasi-shuffle products revisited , J. Algebra 481 , 2017, 293--326

work page 2017
[7]

Ihara, M

K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values , Compositio Math. 142 , 2006, 307--338

work page 2006
[8]

Lascouxand, P

A. Lascouxand, P. Pragacz: Ribbon Schur functions , European J. Combin. 9 , 1988, 561--574

work page 1988

Show all 10 references

[9]

Macdonald: Symmetric functions and Hall polynomials , Oxford University Press

I.G. Macdonald: Symmetric functions and Hall polynomials , Oxford University Press
[10]

Nakasuji, O

M. Nakasuji, O. Phuksuwan and Y. Yamasaki, On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions , Adv. in Math. 333 , 2018, 570--619

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bachmann: Interpolated Schur multiple zeta values , J

H. Bachmann: Interpolated Schur multiple zeta values , J. Aust. Math. Soc., 104 , 2018, 289--307

work page 2018

[2] [2]

Bachmann, Y

H. Bachmann, Y. Yamasaki: Checkerboard style Schur multiple zeta values and odd single zeta values , Math. Zeitschrift, 290 (3), 2018, 1173--1197

work page 2018

[3] [3]

Gessel, G

I. Gessel, G. X. Viennot: Determinants, paths, and plane partitions , 1989, preprint

work page 1989

[4] [4]

G. Z. Giambelli: Alcune proprieta dele funzioni simmetriche caratteristiche , Atti Torino 38 , 1903,823--844

work page 1903

[5] [5]

Hamel, I

A.M. Hamel, I. Goulden: Planar decompositions of tableaux and Schur function determinants , European J. Combin. 16 no. 5, 1995, 461--477

work page 1995

[6] [6]

Hoffman, K

M. Hoffman, K. Ihara: Quasi-shuffle products revisited , J. Algebra 481 , 2017, 293--326

work page 2017

[7] [7]

Ihara, M

K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values , Compositio Math. 142 , 2006, 307--338

work page 2006

[8] [8]

Lascouxand, P

A. Lascouxand, P. Pragacz: Ribbon Schur functions , European J. Combin. 9 , 1988, 561--574

work page 1988