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Zero-Free Columns in Character Tables of Symmetric Groups

This is a Lean formalization of two upper bounds on the number of zero-free columns in the character table of a symmetric group.

Main Results

Write D(n) for the number of conjugacy classes of S_n on which no irreducible character vanishes. Both results assume six cited statements from the literature, bundled as ZeroFree.LiteratureInputs.

  • D(n) is at most a constant times n^{3/4}, for every n ≥ 1.
  • For every B > 5/6 there is a C_B > 0 such that D(n) > C_B n^{1/2} (log n)^B holds for at most C X (log X)^{-(1/2)(B - 5/6) + ε} of the integers n ≤ X, for every ε > 0 and every X ≥ 3.

See §Formal Challenge for a formal certificate.

Dependencies

This depends on Mathlib.

Formal Challenge

A formal challenge file certifying that this repository does formalize the results claimed above is located at Comparator/Challenge.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.

This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:

lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.

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, 'i'); if (__m === '*' || __re.test(location.href)) { injectUserscript("// Add copy buttons to all 
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})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '

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Zero-Free Columns in Character Tables of Symmetric Groups
This is a Lean formalization of two upper bounds on the number of zero-free columns in the character table of a symmetric group.
Main Results
Write D(n) for the number of conjugacy classes of S_n on which no irreducible character vanishes. Both results assume six cited statements from the literature, bundled as ZeroFree.LiteratureInputs.
  • D(n) is at most a constant times n^{3/4}, for every n ≥ 1.
  • For every B > 5/6 there is a C_B > 0 such that D(n) > C_B n^{1/2} (log n)^B holds for at most C X (log X)^{-(1/2)(B - 5/6) + ε} of the integers n ≤ X, for every ε > 0 and every X ≥ 3.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results claimed above is located at Comparator/Challenge.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Force GitHub README to respect dark mode\n(function() {\n var style = document.createElement('style');\n style.textContent = '\n .markdown-body {\n color-scheme: dark light;\n }\n .markdown-body pre { background: #161b22 !important; }\n .markdown-body code { background: rgba(110, 118, 129, 0.4) !important; }\n .markdown-body table th, .markdown-body table td { border-color: #30363d !important; }\n .markdown-body img { background: #0d1117; }\n .markdown-body blockquote { border-left-color: #8b949e; }\n .markdown-body hr { border-color: #30363d; }\n ';\n document.head.appendChild(style);\n})();", "GitHub Dark Mode README Fix");
}
} catch(__e) { console.warn('[Userscript:GitHub Dark Mode README Fix]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Zero-Free Columns in Character Tables of Symmetric Groups
This is a Lean formalization of two upper bounds on the number of zero-free columns in the character table of a symmetric group.
Main Results
Write D(n) for the number of conjugacy classes of S_n on which no irreducible character vanishes. Both results assume six cited statements from the literature, bundled as ZeroFree.LiteratureInputs.
  • D(n) is at most a constant times n^{3/4}, for every n ≥ 1.
  • For every B > 5/6 there is a C_B > 0 such that D(n) > C_B n^{1/2} (log n)^B holds for at most C X (log X)^{-(1/2)(B - 5/6) + ε} of the integers n ≤ X, for every ε > 0 and every X ≥ 3.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results claimed above is located at Comparator/Challenge.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Highlight search terms from Google/DuckDuckGo/Bing referrer\n(function() {\n var ref = document.referrer;\n var terms = [];\n \n if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {\n var url = new URL(ref);\n var q = url.searchParams.get('q') || url.searchParams.get('p');\n if (q) {\n terms = q.split(/\\s+/).filter(function(t) { return t.length > 2; });\n }\n }\n \n if (terms.length === 0) return;\n \n var style = document.createElement('style');\n style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';\n document.head.appendChild(style);\n \n function highlight(node) {\n if (node.nodeType === 3) { // text node\n var text = node.textContent;\n var found = false;\n terms.forEach(function(term) {\n var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\\]\\\\]/g, '\\\\') + ')', 'gi');\n if (regex.test(text)) {\n found = true;\n var frag = document.createDocumentFragment();\n var parts = text.split(regex);\n parts.forEach(function(part, i) {\n if (i % 2 === 0) {\n frag.appendChild(document.createTextNode(part));\n } else {\n var span = document.createElement('span');\n span.className = 'userscript-highlight';\n span.textContent = part;\n frag.appendChild(span);\n }\n });\n node.parentNode.replaceChild(frag, node);\n }\n });\n } else if (node.nodeType === 1 && node.childNodes) { // element\n var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];\n if (!skipTags.includes(node.tagName)) {\n Array.from(node.childNodes).forEach(highlight);\n }\n }\n }\n \n highlight(document.body);\n \n // Re-highlight on dynamic content\n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1 || node.nodeType === 3) highlight(node);\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Highlight Search Terms");
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Zero-Free Columns in Character Tables of Symmetric Groups
This is a Lean formalization of two upper bounds on the number of zero-free columns in the character table of a symmetric group.
Main Results
Write D(n) for the number of conjugacy classes of S_n on which no irreducible character vanishes. Both results assume six cited statements from the literature, bundled as ZeroFree.LiteratureInputs.
  • D(n) is at most a constant times n^{3/4}, for every n ≥ 1.
  • For every B > 5/6 there is a C_B > 0 such that D(n) > C_B n^{1/2} (log n)^B holds for at most C X (log X)^{-(1/2)(B - 5/6) + ε} of the integers n ≤ X, for every ε > 0 and every X ≥ 3.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results claimed above is located at Comparator/Challenge.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Strip utm_, fbclid, gclid, etc. from all links on page\n(function() {\n var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',\n 'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',\n 'ref', 'ref_src', 'source', 'medium', 'campaign'];\n \n function cleanUrl(url) {\n try {\n var u = new URL(url, window.location.origin);\n var changed = false;\n trackingParams.forEach(function(p) {\n if (u.searchParams.has(p)) {\n u.searchParams.delete(p);\n changed = true;\n }\n });\n return changed ? u.toString() : url;\n } catch (e) {\n return url;\n }\n }\n \n function cleanLinks() {\n document.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n \n cleanLinks();\n \n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1) {\n if (node.tagName === 'A') cleanLinks();\n node.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Remove Tracking Parameters from Links");
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '

Skip to content
BranchesTags
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Zero-Free Columns in Character Tables of Symmetric Groups
This is a Lean formalization of two upper bounds on the number of zero-free columns in the character table of a symmetric group.
Main Results
Write D(n) for the number of conjugacy classes of S_n on which no irreducible character vanishes. Both results assume six cited statements from the literature, bundled as ZeroFree.LiteratureInputs.
  • D(n) is at most a constant times n^{3/4}, for every n ≥ 1.
  • For every B > 5/6 there is a C_B > 0 such that D(n) > C_B n^{1/2} (log n)^B holds for at most C X (log X)^{-(1/2)(B - 5/6) + ε} of the integers n ≤ X, for every ε > 0 and every X ≥ 3.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results claimed above is located at Comparator/Challenge.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
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0 forks
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Auto-enable theater mode on YouTube\n(function() {\n function tryTheater() {\n var btn = document.querySelector('button[aria-label=\"Theater mode\"], ytd-player #player button[title=\"Theater mode\"]');\n if (btn && !btn.classList.contains('activated')) {\n btn.click();\n }\n }\n \n // Try immediately\n tryTheater();\n \n // Try after navigation (SPA)\n var lastUrl = location.href;\n setInterval(function() {\n if (location.href !== lastUrl) {\n lastUrl = location.href;\n setTimeout(tryTheater, 500);\n }\n }, 1000);\n \n // Also try on player load\n var observer = new MutationObserver(tryTheater);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "YouTube Theater Mode Default");
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} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Zero-Free Columns in Character Tables of Symmetric Groups
This is a Lean formalization of two upper bounds on the number of zero-free columns in the character table of a symmetric group.
Main Results
Write D(n) for the number of conjugacy classes of S_n on which no irreducible character vanishes. Both results assume six cited statements from the literature, bundled as ZeroFree.LiteratureInputs.
  • D(n) is at most a constant times n^{3/4}, for every n ≥ 1.
  • For every B > 5/6 there is a C_B > 0 such that D(n) > C_B n^{1/2} (log n)^B holds for at most C X (log X)^{-(1/2)(B - 5/6) + ε} of the integers n ≤ X, for every ε > 0 and every X ≥ 3.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results claimed above is located at Comparator/Challenge.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Remove or un-stick sticky/fixed headers that block content\n(function() {\n function unstick() {\n document.querySelectorAll('header, nav, [role=\"banner\"], .header, .navbar, .sticky, .fixed-top, [style*=\"position: fixed\"], [style*=\"position:sticky\"]').forEach(function(el) {\n if (el.style.position === 'fixed' || el.style.position === 'sticky' || \n getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {\n el.style.position = 'static';\n el.style.top = 'auto';\n el.style.zIndex = 'auto';\n }\n });\n }\n \n unstick();\n \n var observer = new MutationObserver(unstick);\n observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });\n})();", "Kill Sticky Headers");
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} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Zero-Free Columns in Character Tables of Symmetric Groups
This is a Lean formalization of two upper bounds on the number of zero-free columns in the character table of a symmetric group.
Main Results
Write D(n) for the number of conjugacy classes of S_n on which no irreducible character vanishes. Both results assume six cited statements from the literature, bundled as ZeroFree.LiteratureInputs.
  • D(n) is at most a constant times n^{3/4}, for every n ≥ 1.
  • For every B > 5/6 there is a C_B > 0 such that D(n) > C_B n^{1/2} (log n)^B holds for at most C X (log X)^{-(1/2)(B - 5/6) + ε} of the integers n ≤ X, for every ε > 0 and every X ≥ 3.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results claimed above is located at Comparator/Challenge.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Universal Dark Mode - works on any site\n(function() {\n var enabled = true;\n \n function applyDarkMode() {\n if (!enabled) return;\n \n // Create style element if it doesn't exist\n var style = document.getElementById('universal-dark-mode-style');\n if (!style) {\n style = document.createElement('style');\n style.id = 'universal-dark-mode-style';\n document.head.appendChild(style);\n }\n \n // Dark mode CSS - inverts colors but preserves images/video\n style.textContent = '\n /* Invert everything except media */\n html {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #1a1a2e !important;\n }\n \n /* Restore images, videos, iframes, canvas */\n img, video, iframe, canvas, svg, picture, [style*=\"background-image\"] {\n filter: invert(1) hue-rotate(180deg) !important;\n }\n \n /* Preserve specific elements that should not be inverted */\n .no-dark-mode, .no-dark-mode *,\n [data-theme=\"light\"], [data-theme=\"light\"],\n .ace_editor, .ace_editor *,\n .CodeMirror, .CodeMirror *,\n .monaco-editor, .monaco-editor *,\n .markdown-body pre, .markdown-body pre *,\n .highlight, .highlight *,\n pre code, pre code * {\n filter: none !important;\n }\n \n /* Fix common UI elements */\n .modal, .popup, .dropdown-menu, .tooltip, .popover {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #2d2d44 !important;\n border-color: #444 !important;\n }\n \n /* Scrollbars */\n ::-webkit-scrollbar { background: #1a1a2e !important; }\n ::-webkit-scrollbar-thumb { background: #444 !important; }\n ::-webkit-scrollbar-thumb:hover { background: #555 !important; }\n \n /* Selection */\n ::selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ::-moz-selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ';\n }\n \n function removeDarkMode() {\n var style = document.getElementById('universal-dark-mode-style');\n if (style) style.remove();\n }\n \n // Toggle with Alt+Shift+D\n document.addEventListener('keydown', function(e) {\n if (e.altKey && e.shiftKey && e.key === 'D') {\n e.preventDefault();\n enabled = !enabled;\n if (enabled) {\n applyDarkMode();\n console.log('[Universal Dark Mode] Enabled');\n } else {\n removeDarkMode();\n console.log('[Universal Dark Mode] Disabled');\n }\n }\n });\n \n // Apply on load\n applyDarkMode();\n \n // Re-apply on dynamic content\n var observer = new MutationObserver(function(mutations) {\n if (enabled && !document.getElementById('universal-dark-mode-style')) {\n applyDarkMode();\n }\n });\n observer.observe(document.head, { childList: true });\n \n console.log('[Universal Dark Mode] Loaded - Press Alt+Shift+D to toggle');\n})();", "Universal Dark Mode");
}
} catch(__e) { console.warn('[Userscript:Universal Dark Mode]', __e); }
})();
})();

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Zero-Free Columns in Character Tables of Symmetric Groups
This is a Lean formalization of two upper bounds on the number of zero-free columns in the character table of a symmetric group.
Main Results
Write D(n) for the number of conjugacy classes of S_n on which no irreducible character vanishes. Both results assume six cited statements from the literature, bundled as ZeroFree.LiteratureInputs.
  • D(n) is at most a constant times n^{3/4}, for every n ≥ 1.
  • For every B > 5/6 there is a C_B > 0 such that D(n) > C_B n^{1/2} (log n)^B holds for at most C X (log X)^{-(1/2)(B - 5/6) + ε} of the integers n ≤ X, for every ε > 0 and every X ≥ 3.
See §Formal Challenge for a formal certificate.
Dependencies
This depends on Mathlib.
Formal Challenge
A formal challenge file certifying that this repository does formalize the results claimed above is located at Comparator/Challenge.lean. This file only depends on Mathlib. It contains formal statements of §Main Results with sorry as proof.
This repository can be verified against the formal challenge with the Lean comparator on a Linux machine. First, follow the instructions in https://github.com/leanprover/comparator to install comparator. Then, run the following command:
lake env comparator Comparator/comparator.json

This repository has been locally verified with the comparator.
About
No description, website, or topics provided.
Resources
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