Bounded core partitions and Borel--Weil--Bott

TL;DR

This work recasts the nonvanishing of cohomology groups $H^i(\mathrm{Gr}(k,n),\Omega^j(t))$ as a combinatorial problem about $t$-core partitions fitting in a $ (n-k)\times k$ rectangle, via the Borel--Weil--Bott theorem. It introduces knijt-partitions and a coarsening method to sharpen existence and vanishing criteria, and proves an elementary Nakano vanishing result for $\Omega^j(t)$ when $t>0$. The paper then develops refined bounds using the $t$-boundary, and completely analyzes the extremal cases $i+j=N$ and $N-1$, including a full description in terms of uniform blocks. In the special case $t=3$, it derives explicit arithmetic criteria tying nonvanishing to $3$-core partitions, including a concrete square-determinant condition and explicit parametrizations, thereby providing concrete, testable cohomology vanishing ranges. The results deepen the bridge between Grassmannian cohomology and the combinatorics of $t$-core partitions and lay groundwork for further connections to Griffiths-type residue constructions on Grassmannians.

Abstract

As a consequence of the Borel--Weil--Bott theorem, the nonvanishing of Hodge numbers of line bundles on a Grassmannian can be reinterpreted in terms of the existence of $t$-core partitions inside a rectangle. We sharpen known conditions for the nonvanishing of Hodge numbers of line bundles on a Grassmannian and give an elementary proof of the Nakano vanishing theorem in this setting. Using the theory of $t$-core partitions, we determine when $H^i(\mathrm{Gr}(k,n),Ω^j(t))$ vanishes in the cases $t=3$ and $i+j\geq k(n-k)-1$.

Figures (6)

• Figure 1: A pair of partitions related by the bijection between $3$-core partitions to $2$-bounded partitions. The $3$-interior of $\lambda$ is shaded, and the hook lengths labeled.
• Figure 2: If $\lambda=(8^1,5^3,3^3)$ as above, then $(r_1,r_2,r_3)=(1,3,3)$ and $(s_1,s_2,s_3)=(3,2,3)$.
• Figure 3: If $\lambda$ is the above $9$-core partition where $\varepsilon_\lambda(9)$ has been shaded, then $(\alpha_1,\alpha_2,\alpha_3)=(1,2,4)$ and $(\beta_1,\beta_2,\beta_3)=(1,3,4)$. For example, $s_2'=s_2+s_3=2+1=3$ is the number of columns in the second block of $\varepsilon_\lambda(9)$.
• Figure 4: All knijt-partitions with $k=8$, $n=12$ and $i+j=N$. The $t$-interior of each is shaded with the possible values of $t$ given.
• Figure 5: A knijt-partition $\lambda$ satisfying $i+j=N-1$ for $n=11$ and $i>0$. The only other such partition is $\lambda^\mathsf{T}$. Both require that $i=4$ and $t=6$.

Theorems & Definitions (34)

• Definition 2.1
• Theorem 2.2: Borel--Weil--Bott, A.9 in Nik25
• Remark
• Theorem 2.3
• proof
• Theorem 2.4
• Theorem 3.1
• proof
• Lemma 3.2
• proof