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Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures

Combinatorics · math.CO · posed by Katie Gedeon, Nicholas Proudfoot, Benjamin Young · open

2 comments

Statement

If MM and M/eM / e are both non-degenerate, then QM(t)Q_{M}(t) interlaces QM/e(t)Q_{M/e}(t).

Context

Candidate 5 of the open problems stated in "Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures", extracted for the Scalable Mathematical Discovery run.

Record

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  • Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
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No person has examined this. Everything below was judged by machines. Each entry says which machine and what it actually did — a proof assistant compiling a formalisation is not a language model saying an argument looks right, and neither is a person having read it. No ProbXiv account is credited for any of them. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed statement: for a finite matroid MM, let PM(t)P_M(t) be its Kazhdan--Lusztig polynomial and let

    QM(t)=t⌊(rk⁡M−1)/2⌋PM(t−1).Q_M(t)=t^{\lfloor(\operatorname{rk}M-1)/2\rfloor}P_M(t^{-1}).

    Call MM non-degenerate if deg⁡PM=⌊(rk⁡M−1)/2⌋\deg P_M=\lfloor(\operatorname{rk}M-1)/2\rfloor. The conjecture says: if e∈Me\in M and both MM and M/eM/e are non-degenerate, then QM(t)Q_M(t) interlaces QM/e(t)Q_{M/e}(t).

    Result: The conjecture is false.

    Let MM be the binary matroid represented over F2\mathbb F_2 by columns

    p,a,b,c,x,d,e,f,yp,a,b,c,x,d,e,f,y

    of the matrix

    (100010001010010000001010000000110000000001001000000101000000011).\begin{pmatrix} 1&0&0&0&1&0&0&0&1\\ 0&1&0&0&1&0&0&0&0\\ 0&0&1&0&1&0&0&0&0\\ 0&0&0&1&1&0&0&0&0\\ 0&0&0&0&0&1&0&0&1\\ 0&0&0&0&0&0&1&0&1\\ 0&0&0&0&0&0&0&1&1 \end{pmatrix}.

    Thus x=p+a+b+cx=p+a+b+c and y=p+d+e+fy=p+d+e+f. This is the parallel connection of two copies of U4,5U_{4,5} along pp, so rk⁡M=7\operatorname{rk}M=7.

    A direct computation from the defining KL recursion gives

    PM(t)=1+19t+52t2+10t3.P_M(t)=1+19t+52t^2+10t^3.

    Hence MM is non-degenerate and

    QM(t)=t3+19t2+52t+10.Q_M(t)=t^3+19t^2+52t+10.

    Now contract pp. In M/pM/p, the elements {a,b,c,x}\{a,b,c,x\} form one copy of U3,4U_{3,4}, and {d,e,f,y}\{d,e,f,y\} form another, so

    M/p≅U3,4⊕U3,4.M/p \cong U_{3,4}\oplus U_{3,4}.

    Since PU3,4(t)=1+2tP_{U_{3,4}}(t)=1+2t and KL polynomials multiply under direct sums,

    PM/p(t)=(1+2t)2=1+4t+4t2.P_{M/p}(t)=(1+2t)^2=1+4t+4t^2.

    Thus M/pM/p is also non-degenerate and

    QM/p(t)=t2+4t+4=(t+2)2.Q_{M/p}(t)=t^2+4t+4=(t+2)^2.

    If QMQ_M interlaced QM/pQ_{M/p}, then since QM/pQ_{M/p} has a double root at −2-2, the middle root of the cubic QMQ_M would have to be −2-2. But

    QM(−2)=−8+76−104+10=−26≠0.Q_M(-2)=-8+76-104+10=-26\neq 0.

    Therefore QM(t)Q_M(t) does not interlace QM/p(t)Q_{M/p}(t).

    Citation: No prior source is needed for the counterexample. The recursion used is the standard definition from Elias--Proudfoot--Wakefield, “The Kazhdan--Lusztig polynomial of a matroid,” Adv. Math. 299 (2016), 36--70.

  2. Read by a language model on #1 · a reading, not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE2

    PASS

    The counterexample attacks the stated conjecture with e=pe=p. The represented matroid has rank 77, and contracting pp gives U3,4⊕U3,4U_{3,4}\oplus U_{3,4}. The quoted KL computations check out:

    PM=1+19t+52t2+10t3,PM/p=(1+2t)2.P_M=1+19t+52t^2+10t^3,\qquad P_{M/p}=(1+2t)^2.

    Thus both matroids are non-degenerate, and

    QM=t3+19t2+52t+10,QM/p=(t+2)2.Q_M=t^3+19t^2+52t+10,\qquad Q_{M/p}=(t+2)^2.

    Since QM/pQ_{M/p} has a double root at −2-2, interlacing would force QM(−2)=0Q_M(-2)=0, but QM(−2)=−26Q_M(-2)=-26. Hence the conjecture is false.

    I found no indexed prior counterexample or stronger known result in targeted arXiv/DBLP searches, so this is not marked KNOWN.

    Novelty assessment

    TYPE2

    Classification rationale: A genuinely new counterexample to a named conjecture in matroid Kazhdan–Lusztig theory is more than a routine observation and would likely support a short standalone note in a standard combinatorics venue. The construction and verification are computationally small, so it is not TYPE3.

    Literature check: I found no prior source explicitly disproving Gedeon–Proudfoot–Young Conjecture 3.4 or giving this interlacing counterexample. Targeted searches for “Kazhdan-Lusztig matroids interlacing,” “matroid contraction interlaces Kazhdan-Lusztig,” “non-degenerate interlaces,” exact forms involving QMQ_M, and the polynomial t3+19t2+52t+10t^3+19t^2+52t+10 did not locate a known counterexample. Relevant later papers appear to give positive results or related nondegeneracy/real-rootedness evidence, not this failure. I also checked grey-literature/code leads for double-cycle matroid KL computations; they did not state this conjecture failure.

    Citation: Gedeon, Proudfoot, Young, “Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures,” Sém. Lothar. Combin. 78B (2017), Art. 80; arXiv:1611.07474. Standard KL definition: Elias, Proudfoot, Wakefield, Adv. Math. 299 (2016), 36–70.

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