Pith. sign in

REVIEW 2 major objections 3 minor 43 references

Factorization of Schur polynomials twisted by roots of unity and a reciprocal pair

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Schur polynomials at roots of unity plus a reciprocal pair evaluate to a signed product of three hyperbolic factors, so every partition is compressed to three integers and a sign.

desk verdict Solid new evaluation theorem with a complete proof; the abstract's 'nothing else' minimality claim outruns what is proved and should be fixed in revision. read the letter →

arxiv 2608.09619 v1 pith:NUFNL3PA submitted 2026-08-10 math.CO math.RT

Carles Marín This is my paper · ORCID
classification math.COmath.RT MSC 05E0505A1505E1020G05
keywords Schurpolynomialsrootsofunityreciprocalvariablesfactorizationcoresandquotientsbetasetssignedenumeration
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 establishes that the Schur polynomial of any partition $\lambda$, evaluated at the alphabet $\{1,\zeta,\ldots,\zeta^{t-1},z,z^{-1}\}$ consisting of all $t$-th roots of unity plus one free reciprocal pair, is either zero or a fixed product of three hyperbolic sine factors divided by a fixed denominator. The three arguments and the overall sign are read directly from the residue profile of $\lambda$'s $\beta$ set, so the value depends on a multiset of three integers and a sign, and on nothing else. This means partitions of unrelated sizes can share one value, and the paper writes down the infinite fibers of this collapse explicitly. A vanishing criterion follows: the value is zero exactly when a residue class modulo $t$ is empty, or (only for even $t$) when the two distinguished $\beta$-intervals are concentric. The result also yields an extension of the independence criterion for two-row shapes and, at $t=2$, a $(-1)$-enumeration of plane partitions refined by a free parameter.

What carries the argument

The proof runs through the bialternant formula for Schur polynomials. Expanding the numerator by Laplace along the $t$ 'frozen' rows that carry the root-of-unity variables leaves one term for each choice of two un-frozen columns; the fact that there are $t+2$ columns in total and $t$ residue classes forces a trichotomy of residue profiles, and a single cancellation lemma in the symmetric group collapses the surviving terms into the three-factor product. The geometric reading used throughout identifies $d_1,d_2$ as the lengths of two intervals on the beta line and $d_3$ as twice the distance between their centres; this 'interval triple' is what the evaluation sees, and its symmetry is the reason the value records a multiset rather than an ordered triple.

What would settle it

For $t=7$, enumerate all partitions of size at most $20$ and compare the exact bialternant with the right-hand side of (5), then search for pairs with the same multiset $\{d_1,d_2,d_3\}$ and same sign whose exact values differ; either search can falsify the paper's claims at the first counterexample.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.1: for $t\ge 2$ and $\lambda$ a partition with at most $t+2$ parts, with $\beta(\lambda)$ the shifted $\beta$ set and $n_i(\lambda)$ the counts of $\beta$ parts congruent to $i$ modulo $t$, the value $\Phi_t(\lambda;z)=s_\lambda(1,\zeta,\ldots,\zeta^{t-1},z,z^{-1})$ vanishes if any $n_i=0$; otherwise the residue profile is either 'two-class' (two residue classes each contribute two $\beta$ parts) or 'size-three' (one class contributes three), and in both cases one has $$\Phi_t(\$\lambda$;z)=\varepsilon_\$\lambda$\,\frac{\$\sinh$(d_1\$\theta$/2)\$\sinh$(d_2\$\theta$/2)\$\sinh$(d_3\$\theta$/2)}{\$\sinh$^2(t\$\theta$/2)\$\sinh$\$\theta$},\quad z=e^\$\theta$,$$ where $d_1,d_2$ are the gaps inside the two distinguished classes, $d_3$ is the distance between their sums, and $\varepsilon_\lambda=\pm1$ is an explicit sorting sign. The paper proves this for every $\lambda$ with no hypothesis on its shape, and shows that the formula is symmetric in $d_1,d_2,d_3$, so the actual invariant is the multiset $\{d_1,d_2,d_3\}$ plus the sign. Since the right-hand side is a Laurent polynomial in $z$, the theorem also gives, via $\mathfrak{sl}_2$ characters, a uniform character-ratio form with no exponentials.

Load-bearing premise

The load-bearing premise is that the evaluation invariant (multiset of three integers plus sign) captures all the information in the value for every $t$; the formula is proved for all $t$, but the 'nothing else' part is only checked for $t\le6$, so the compression claim for larger $t$ rests on an unproved minimality statement.

Editorial extensions

If this is right

  • For any $t\ge2$, computing $s_\lambda$ at this alphabet reduces to reading three integers and a sign off the $t$-residue profile; no expansion of $\lambda$ is needed.
  • The value vanishes exactly when a residue class modulo $t$ is empty, or when the two distinguished intervals are concentric (the latter possible only for even $t$).
  • For two-row shapes, the value equals $\pm s_\lambda(z,z^{-1})$ not only on the $t$-cores but on one additional family classified by its core and quotient; this is the full correction to the independence criterion on the reciprocal locus.
  • At $t=2$, the theorem gives a product formula for a $(-1)$-weighted count of plane partitions in a box, refined by a free parameter $z$; at $z=1$ it recovers signed counts such as $(c/2+1)^2$ for $2\times2\times c$ boxes with $c$ even and $0$ for $c$ odd.
  • The factorization is isolated: adding a second reciprocal pair, enlarging the root-of-unity orbit, replacing the orbit by a coset, or replacing the reciprocal pair by a free pair all destroy the product; only the zero locus survives for arbitrary numbers of pairs, with two explicit conditions that are proved sufficient and, in the proved ranges, necessary.

Reading between the lines

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

  • If the compression holds for all $t$, the same evaluation invariant may govern other specializations that adjoin exactly one free direction to a full orbit, suggesting a general rank-one evaluation theorem for characters of classical type.
  • The counting argument that three factors are forced by translation invariance of an interval pair indicates that any alphabet with excess two, for instance in flagged or skew settings, should exhibit a three-factor product independently of the Laplace-expansion proof.
  • The $t=2$ signed enumeration with a free parameter invites a cyclic-sieving refinement: a $q$-analogue of $\Phi_t$ with a cyclic action whose fixed points are counted by the refined signed count; the paper leaves this as an open problem.
  • The determinant dichotomy of Section 8, where alphabet determinant $+1$ gives factorization on self-complementary shapes while determinant $-1$ gives vanishing, may transfer to other groups and other order-two fixed letters, providing a test for universal-character analogues.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies the Schur polynomial s_λ(1, ζ, ..., ζ^{t-1}, z, z^{-1}) for t ≥ 2 and arbitrary partitions λ with at most t+2 parts. Theorem 3.1 gives a closed form: the value is zero if some residue class modulo t is empty, and otherwise equals a signed product of three hyperbolic sine factors divided by sinh^2(tθ/2) sinh θ, with the three integers d_1, d_2, d_3 and the sign read off the beta set of λ. The proof is a Laplace expansion along the t frozen rows, with explicit lemmas for the Vandermonde minors, the column move, and the cancellation identities. The paper derives several consequences: a geometric vanishing criterion, an extension of an independence criterion of Ayyer–Kumari for two-row shapes, a signed enumeration of plane partitions at t = 2 refined by a free parameter, a discussion of four deformations that destroy the factorization, and a partly conjectural determination of the zero locus of Ψ_r = s_λ(1, -1, z_1^{±1}, ..., z_r^{±1}). The main evaluation theorem is supported by 10,959 exact numerical checks, and all computational claims are documented in an ancillary archive.

Significance. If the main claims hold, Theorem 3.1 is a clean and remarkably complete evaluation for this mixed alphabet, with an explicit sign and no hypothesis on the shape of λ. The proof is detailed and appears sound: it gives a genuine Laplace-expansion argument, explicit lemmas, and a derived (not fitted) sign formula. The paper also provides reproducible code and archived outputs, and it carefully labels conjectural versus proved statements, which is exemplary. The zero-locus section is more speculative but honestly framed. However, the headline compression claim that the value sees 'a multiset of three integers and a sign, and nothing else' is not proved in the printed text; it is verified only for finite ranges. Since this minimality claim is central to the abstract's 'determine exactly how much' assertion, the paper needs either a proof of that claim or a corresponding weakening of the claim.

major comments (2)
  1. [Abstract; §3.4; §9] The assertion that Φ_t(λ;z) depends on a multiset of three integers and a sign 'and nothing else' is not established by Theorem 3.1. Theorem 3.1 proves that the value factors through the ordered invariant I_t = (d_1, d_2, d_3, ε_λ) defined in (7); symmetry of the numerator in (5) then reduces the ordered triple to a multiset. But the converse direction — that two distinct multisets cannot produce the same rational function — is only checked computationally: §3.4 reports the absence of collisions only over |λ| ≤ 20 for t = 3, 4, and the verification table in §9 contains no row proving minimality for all t. This is not a cosmetic gap, because each factor u^{d_i} − u^{−d_i} in the numerator is reducible (u^d − u^{−d} = u^{−d} ∏_{m|2d} Φ_m(u)), so additive identities among divisor multisets could in principle make distinct triples coincide. The paper should either prove the minimality claim, for example by a cyclotomic-factor argument in the spirit of Lemma 5.1, or explicitly weaken the abstract and §3.4 to the factorization statement that is actually proved.
  2. [§8.4, sentence after Lemma 8.9] The claim that the converse of Theorem 8.1 holds for every r when |λ| ≤ 2r + 2 outside Littlewood's range is asserted in a single sentence: 'there Littlewood's rule applies verbatim and the converse follows by the argument of Theorem 8.4.' That is not demonstrated. The proof of Theorem 8.4 uses ℓ(λ) ≤ N/2 in several essential places: it uses the β' = ∅ term to get m_μ ≥ 1 for every μ ⊆ λ, it uses ℓ(μ*) > ℓ(λ) when μ = λ, and it uses horizontal-strip constructions that depend on the row structure. When ℓ(λ) > N/2, none of these steps is automatic, and the text gives no witness construction for the unstable band. Since the abstract explicitly claims this converse as proved, the argument needs to be supplied, or the claim should be moved to the conjectural part of Section 8.
minor comments (3)
  1. [§8.2, Eq. (21)] In the displayed identity (21), the exponent s in (-1)^s is undefined. From the preceding expression it should be the parity of \binom{N}{2} + r (or the sum itself), and this should be stated explicitly.
  2. [§9, verification table] The row labelled 'Theorem 8.1, both directions' is misleading: Theorem 8.1 as stated contains only the sufficient direction for all r, while the converse for all r is Conjecture 8.6. The numerical verification of the converse over r ≤ 3 should be labelled as a check of the conjecture, not of the theorem.
  3. [§4, proof of Lemma 4.4] The parity computation in the proof of Lemma 4.4 is hard to audit as printed, especially the step that obtains κ_{1j}/κ_{2j} = -1 from the exponent '2j_{A2} − 1' and the indicator [a_2 < b_j < a_1]. Please expand that display into explicit parity bookkeeping, since the lemma is load-bearing for the sign in Theorem 3.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.1 is derived from the bialternant by Laplace expansion; the unproved minimality claim is a completeness gap, not a circular input.

full rationale

The paper's central derivation starts from the bialternant definition of Schur polynomials and from external classical evaluations (Littlewood–Richardson [LR34], Ayyer–Kumari [AK22, AK25]). Theorem 3.1's product formula (5) is obtained by a Laplace expansion along the t frozen rows with a cancellation lemma in the symmetric group; the sign epsilon_lambda is derived in (6) and Proposition 3.10, not fitted to data. No parameter in (5) is calibrated to make the identity true: d1, d2, d3 are read off the beta set, and the formula is independently checked over 10,959 exact cases in Section 9. The compression claim in the abstract, that the value sees 'a multiset of three integers and a sign, and nothing else,' has a genuinely unproved minimality component: the text after Proposition 3.10 says 'that part remains a check rather than a proof,' verified only over t <= 6 and |lambda| <= 14. That is a limitation of the converse direction of the minimality statement, not circularity, because the evaluation formula is not defined in terms of the triple and no equality between distinct triples is assumed by construction. The conjectural parts, including Conjecture 8.6 and the minimality of the multiset, are explicitly labelled as conjectures or checks. Self-citation is limited to [Mar26] in Remark 8.11, where it is contextual (the origin of the alphabet) and not load-bearing for Theorem 3.1, Theorem 5.2, or the zero-locus theorems. The one place where the proof depends on cited results, Lemma 8.8 and the reduction in Section 8.4, cites Ayyer–Kumari's universal character identities and independent symplectic modification rules, not the author's own prior work. No step of the derivation reduces to its own input by construction, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters or invented entities appear. The only inputs are the structural integer t, the partition λ, and the free variable z; the sign and the three distances are explicit formulas in β(λ). The ledger records the cited background results the proof leans on, all standard in symmetric function theory and representation theory.

assumptions (6)
  • standard math Bialternant formula for Schur polynomials: s_λ = det(x_i^{β_j}) / det(x_i^{N-j}).
    Used throughout, starting from Lemma 4.3 and the proof of Theorem 3.1.
  • standard math Littlewood-Richardson evaluation: s_λ(1,ζ,...,ζ^{t-1}) is 0 or ±sgn(σ) depending on whether the t-core is empty.
    Quoted as Theorem 2.1 from [LR34]; used in the splitting identity (18) and as context.
  • standard math Core-quotient bijection and Garvan-Kim-Stanton coordinates identify t-cores with root lattice vectors of type A_{t-1}.
    Used in Propositions 3.4, 3.5, 3.6 and Remark 8.7.
  • standard math Macdonald's skew form of Littlewood's theorem: s_{λ/ν}(μ_t) vanishes unless λ/ν is tileable by t-ribbons and is otherwise a ribbon sign times a quotient product.
    Used for the signed enumeration in Section 6 and for the splitting identity (18).
  • standard math Koike-Terada universal characters o_ν, sp_ν and their relations, including the reduction o_ν(W,1,-1)=sp_ν(W) via [AK25].
    Used in Lemma 8.8 and the Section 8.4 reduction to symplectic characters.
  • standard math Littlewood's restriction rule m_ν(λ)=Σ_{β' even} c^λ_{νβ'} holds for ℓ(λ)≤N/2.
    Used in the proof of Theorem 8.4; the paper explicitly notes the rule stops applying outside this range.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Factorization of Schur polynomials twisted by roots of unity and a reciprocal pair." pith.science (2026). https://pith.science/paper/NUFNL3PA

@misc{pith2026260809619,
  author       = {Pith},
  title        = {Pith review of: Factorization of Schur polynomials twisted by roots of unity and a reciprocal pair},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NUFNL3PA}},
  note         = {Machine review of arXiv:2608.09619}
}
abstract

Let $\mu_t$ be the full set of $t$-th roots of unity and $(z,z^{-1})$ a free reciprocal pair. We determine how much of a partition $s_\lambda(\mu_t,z,z^{-1})$ can see: a multiset of three integers and a sign, and nothing else, so partitions of any sizes agreeing on it share the value. The evaluation, for every $t\ge2$ and every $\lambda$ with no hypothesis on its shape, is a signed product of exactly three factors over a fixed denominator, or zero, the three arguments read off the $t$-quotient. The proof is a Laplace expansion along the $t$ frozen rows of the bialternant with one cancellation lemma in the symmetric group, and delivers the sign, the one already in Littlewood's evaluation at $\mu_t$. Three consequences follow. A vanishing criterion: it vanishes exactly when a residue class modulo $t$ is empty, or two distinguished classes are concentric as intervals, the second only for $t$ even. An extension of a recent independence criterion of Ayyer-Kumari: for two-row shapes on the reciprocal locus it acquires exactly one further family, classified by core and quotient. And an enumerative reading: at $t=2$ a $(-1)$-enumeration of plane partitions in a box refined by a parameter that stays free. The factorization is isolated: it fails under each of four deformations of the alphabet, for one reason. A last section treats the zero locus, which survives further pairs. Two conditions make $\Psi_r=s_\lambda(1,-1,z_1^{\pm1},\dots,z_r^{\pm1})$ vanish: the beta set having constant parity, and $\lambda$ being self-complementary of odd width. That direction is a corollary of the complementation identity over an index family Ayyer and Behrend single out. The new content is the converse, that nothing else vanishes, proved for one pair, for every $r$ inside Littlewood's range, and for every $r$ when $|\lambda|\le2r+2$. The rest is conjectural, verified over every shape in the tabulated ranges.

Figures

Figures reproduced from arXiv: 2608.09619 by the authors.

Figure 1
Figure 1. The mechanism for t = 3, λ = (3, 2): here N = 5, β = (7, 5, 2, 1, 0) with residues (1, 2, 2, 1, 0) modulo 3, so no class is empty and two classes have size two. The three arguments of (5) are the two interval lengths and twice the distance between the interval centres; the short double arrow marks that distance, here 1 2 , so that d3 = 1. mi are invisible to (d1, d2, d3) by Proposition 3.4, and it would be natural t… view at source ↗
Figure 2
Figure 2. The image of λ 7→ (d1, d2, d3) over all λ with |λ| ≤ 20 and at most t + 2 rows, for t = 3 (left) and t = 4 (right), taken up to the exchange of d1 and d2, under which (5) is symmetric; without that identification, and with A and B in the order rA ≤ rB of Theorem 3.1, the two panels carry 126 and 92 points rather than 96 and 73. The compression is the point of the picture: 860 partitions land on 96 triples on the lef… view at source ↗
Figure 3
Figure 3. The fibres of the evaluation invariant (7), the complement of [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: is the statement computed rather than asserted: both families are plotted over a range of t, and the odd t where (16) cannot occur is drawn as the control. 0 1 2 3 4 2 0 1 2 3 4 5 6 1 t = 3 (odd: no extra family) t-cores (6) extra family (0) 0 1 2 3 4 5 2 0 2 4 6 8 10 …
Figure 5
Figure 5. Figure 5: The signed counts of Example 6.2: the value of Corollary 6.1 at z = 1 and t = 2 for the rectangular shapes λ = (c r ), that is the (−1)-enumeration of plane partitions in an r × (4 − r) × c box. Every entry is the closed form ελ d1d2d3/(2t 2 ), cross-checked against a …
Figure 6
Figure 6. Figure 6: Proposition 6.4, drawn. For each k the admissible ν are the single family (m + k, m), and σm = sλ/(m+k,m)(1, −1) is plotted against m and k; the grey dots are the zeros. Every bar has height exactly ±1, which is part (i): the terms are a set, so the cancellation the en…
Figure 7
Figure 7. Figure 7: The zero locus of Theorem 8.1, counted rather than plotted. For each size |λ| the bars give the number of partitions with at most N parts that vanish, split by branch; the shaded step is the total number of shapes of that size, on the same axis, so that the rarity of t…
Figure 8
Figure 8. Figure 8: Why self-complementarity kills the character. Expanding along the two frozen rows leaves terms indexed by pairs (e, o) with e ∈ E, o ∈ O, and the reflection x 7→ c−x acts on them. Above, a self-complementary shape of odd width: every arc lands in β and joins entries of…
Figure 9
Figure 9. Figure 9: The reduction of Lemma 8.8, drawn. Each label ν occurring in sλ = ν cνoν is placed in one of four classes and its value oν(A) computed. Self-associate labels vanish, every one of them; labels standard with ν ′ 1 > N/2 fold onto the basis as ±oµ(A); non-standard labels …
Figure 10
Figure 10. Figure 10: Where one reciprocal pair stops being typical. Each partition with at most N parts is a point at (|λ|, ℓ(λ)); the class drawn as a star is the one that vanishes at the endpoint zi = 1 without vanishing identically, and it is the whole content of the picture; each pane…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 40 canonical work pages

  1. [1]

    S. P. Albion, Universal characters twisted by roots of unity, Algebraic Combinatorics; arXiv:2212.07343

    arXiv

  2. [2]

    S. P. Albion, Character factorisations, z -asymmetric partitions and plethysm, arXiv:2501.18520

    work page Pith review arXiv

  3. [3]

    The Cone of Cyclic Sieving Phenomena

    P. Alexandersson and N. Amini, The cone of cyclic sieving phenomena, Discrete Math. 342 (2019), no. 6, 1581--1601; arXiv:1804.01447

    work page Pith review arXiv 2019

  4. [4]

    Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

    A. Ayyer and R. E. Behrend, Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions, J. Combin. Theory Ser. A 165 (2019), 78--105; arXiv:1804.04514

    work page Pith review arXiv 2019

  5. [5]

    Factorization of classical characters twisted by roots of unity

    A. Ayyer and N. Kumari, Factorization of classical characters twisted by roots of unity, J. Algebra 609 (2022), 437--483; arXiv:2109.11310

    work page Pith review arXiv 2022

  6. [6]

    Ayyer and N

    A. Ayyer and N. Kumari, Further results for classical and universal characters twisted by roots of unity, J. Algebraic Combin. 63 (2025), no. 1; arXiv:2501.00275

    arXiv 2025

  7. [7]

    Ciucu and C

    M. Ciucu and C. Krattenthaler, A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings, in: Advances in Combinatorial Mathematics, Springer, Berlin, 2009

    work page 2009

  8. [8]

    Garvan, D

    F. Garvan, D. Kim and D. Stanton, Cranks and t -cores, Invent. Math. 101 (1990), no. 1, 1--18

    work page 1990

Show all 43 references
  1. [9]

    Hidaka and M

    M. Hidaka and M. Itoh, The Schur polynomials in all primitive n th roots of unity, J. Combin. Theory Ser. A; arXiv:2403.10817

  2. [10]

    J. C. Jantzen, Darstellungen halbeinfacher algebraischer Gruppen, Bonner Math. Schriften 67 (1973)

    1973

  3. [11]

    Karmakar, Character factorizations for representations of GL(n, ) , arXiv:2210.03544

    C. Karmakar, Character factorizations for representations of GL(n, ) , arXiv:2210.03544

    arXiv

  4. [12]

    Karmakar, Character values at elements of order two, arXiv:2412.17324

    C. Karmakar, Character values at elements of order two, arXiv:2412.17324

    arXiv

  5. [13]

    Mar\'in, Schur functions at (1,-1,t,t^ -1 ) : a closed form on the non-identity component of O(4) , preprint, 2026; doi:10.5281/zenodo.21463000

    C. Mar\'in, Schur functions at (1,-1,t,t^ -1 ) : a closed form on the non-identity component of O(4) , preprint, 2026; doi:10.5281/zenodo.21463000

    2026 doi

  6. [14]

    Z. Jin, N. Jing, Z. Li and D. Wang, Universal symplectic/orthogonal functions and general branching rules, arXiv:2209.00767

    arXiv

  7. [15]

    Kumar, G

    S. Kumar, G. Lusztig and D. Prasad, Characters of simply-laced nonconnected groups versus characters of nonsimply-laced connected groups, Contemp. Math. 478 (2009), 99--101; arXiv:math/0701615

    2009 arXiv

  8. [16]

    Knutson and T

    A. Knutson and T. Tao, The honeycomb model of GL_n( ) tensor products I: proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055--1090

    1999

  9. [17]

    Kumari, Factorization of classical characters twisted by roots of unity: II, J

    N. Kumari, Factorization of classical characters twisted by roots of unity: II, J. Pure Appl. Algebra 228 (2024), no. 11, 107714; arXiv:2212.12477

    2024 arXiv

  10. [18]

    Kumari, Characters of classical groups twisted by roots of unity, PhD thesis, Indian Institute of Science, Bangalore, 2023

    N. Kumari, Characters of classical groups twisted by roots of unity, PhD thesis, Indian Institute of Science, Bangalore, 2023

    2023

  11. [19]

    Lecouvey, Parabolic Kazhdan--Lusztig polynomials, plethysm and generalized Hall--Littlewood functions for classical types, European J

    C. Lecouvey, Parabolic Kazhdan--Lusztig polynomials, plethysm and generalized Hall--Littlewood functions for classical types, European J. Combin. 30 (2009), no. 1, 157--191

    2009

  12. [20]

    D. E. Littlewood and A. R. Richardson, Immanants of some special matrices, Q. J. Math. os-5 (1934), no. 1, 269--282

    1934

  13. [21]

    Lascoux, B

    A. Lascoux, B. Leclerc and J.-Y. Thibon, Ribbon tableaux, Hall--Littlewood functions, quantum affine algebras and unipotent varieties, J. Math. Phys. 38 (1997), no. 2, 1041--1068

    1997

  14. [22]

    Lee and Y.-T

    S.-Y. Lee and Y.-T. Oh, Skew Schur polynomials and cyclic sieving phenomenon, Electron. J. Combin. 29 (2022), no. 4, Paper 4.6; arXiv:2112.12394

    2022 arXiv

  15. [23]

    I. G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford University Press, 1995

    1995

  16. [24]

    Reiner, D

    V. Reiner, D. Stanton and D. White, The cyclic sieving phenomenon, J. Combin. Theory Ser. A 108 (2004), no. 1, 17--50

    2004

  17. [25]

    J. R. Stembridge, On minuscule representations, plane partitions and involutions in complex Lie groups, Duke Math. J. 73 (1994), no. 2, 469--490

    1994

  18. [26]

    Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, J

    S. Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra 205 (1998), no. 2, 337--367

    1998

  19. [27]

    Prasad, A character relationship on GL_n( ) , Israel J

    D. Prasad, A character relationship on GL_n( ) , Israel J. Math. 211 (2016), no. 1, 257--270

    2016

  20. [28]

    Ayyer and I

    A. Ayyer and I. Fischer, Bijective proofs of skew Schur polynomial factorizations, J. Combin. Theory Ser. A 174 (2020), 105241; arXiv:1905.05226

    2020 arXiv

  21. [29]

    Eisenk\"olbl, A Schur function identity related to the (-1) -enumeration of self-complementary plane partitions, J

    T. Eisenk\"olbl, A Schur function identity related to the (-1) -enumeration of self-complementary plane partitions, J. Combin. Theory Ser. A 115 (2008), 199--212; arXiv:math/0602401

    2008 arXiv

  22. [30]

    Fauser, P

    B. Fauser, P. D. Jarvis, R. C. King and B. G. Wybourne, New branching rules induced by plethysm, J. Phys. A 39 (2006), 2611--2655; arXiv:math-ph/0508034

    2006 arXiv

  23. [31]

    Gavarini, A Brauer algebra-theoretic proof of Littlewood's restriction rules, J

    F. Gavarini, A Brauer algebra-theoretic proof of Littlewood's restriction rules, J. Algebra 212 (1999), 240--271

    1999

  24. [32]

    T. J. Enright and J. F. Willenbring, Hilbert series, Howe duality and branching for classical groups, Ann. of Math. 159 (2004), 337--375

    2004

  25. [33]

    M. J. Newell, Modification rules for the orthogonal and symplectic groups, Proc. Roy. Irish Acad. Sect. A 54 (1951), 153--163

    1951

  26. [34]

    Sundaram, Tableaux in the representation theory of the classical Lie groups, in Invariant Theory and Tableaux (Minneapolis, MN, 1988), IMA Vol

    S. Sundaram, Tableaux in the representation theory of the classical Lie groups, in Invariant Theory and Tableaux (Minneapolis, MN, 1988), IMA Vol. Math. Appl. 19, 191--225, Springer-Verlag, New York, 1990

    1988

  27. [35]

    Ishikawa, S

    M. Ishikawa, S. Okada, H. Tagawa and J. Zeng, Generalizations of Cauchy's determinant and Schur's Pfaffian, Adv. in Appl. Math. 36 (2006), 251--287

    2006

  28. [36]

    R. C. King, Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups, J. Math. Phys. 12 (1971), 1588--1598

    1971

  29. [37]

    Koike and I

    K. Koike and I. Terada, Young-diagrammatic methods for the representation theory of the classical groups of type B_n , C_n , D_n , J. Algebra 107 (1987), 466--511

    1987

  30. [38]

    Bunder and J

    For the name and the surrounding family of identities (Cassini, Catalan, d'Ocagne, Vajda) see e.g.\ M. Bunder and J. Tonien, Cassini, d'Ocagne and Vajda identities for n -step Fibonacci numbers, arXiv:2201.06269

    arXiv

  31. [39]

    Kumari and A

    N. Kumari and A. Stokke, Murnaghan--Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions, Discrete Math. 349 (2026), no. 2; arXiv:2411.11447

    2026 arXiv

  32. [40]

    Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Ann

    G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Ann. of Math. (2) 156 (2002), 835--866

    2002

  33. [41]

    D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, 2nd ed., Oxford University Press, 1950

    1950

  34. [42]

    Sathish Kumar, Flagged skew Schur polynomials twisted by roots of unity, arXiv:2306.10289

    V. Sathish Kumar, Flagged skew Schur polynomials twisted by roots of unity, arXiv:2306.10289

    arXiv

  35. [43]

    R. P. Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), 103--113

    1986

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.